_{This summarises to my 3options Answer, which deals specifically with the creatures you cite as examples in your Question, but below that there's some additional material on the topic.}

SUMMARY

The Greek deities did still show up and participate, in person, in wars among humans in the time between Homer and Herodotus. It was a much rarer occurrence than in the archaic Age of Heroes but it did happen, at least in the popular imagination of later times.

There were entire tribes of fantastical peoples and an assortment of fabulous animals detailed by Greek and Roman writers. These writers had so much of the respect of Western Europeans in the so-called Late Antique and Mediaeval eras that numerous such creatures (unicorns, e.g.) were actively believed in by Mediaeval Western Europeans and are still popular in fantasy fiction and children’s cartoons today, even though it might be mostly unknown to the consumers of these art-forms that significant aspects of their material originate in Ancient Greek folklore.

The appearances of Dionysos [Dionysus] and Artemis among mortals in the contexts that you’ve described were not mundane sightings to which merely random dudes were privy on a regular basis or anything close to it. Dionysos’ adventures among mortals almost invariably take place before he becomes a full-fledged Olympian god, in the earliest period of his life while he is still roaming the earth, collecting followers, converting hapless cities to his religion and conquering his relatives in India. Once he does this, he gains the full respect of his fellow deities and joins their ranks, after which his adventures and wars are mostly concluded.

Artemis was in the habit of hunting in wilderness areas of mountains and forests, away from human interference. Her most famous hunting story is, to me, evidently an illustration of this. Her own great-nephew the hunter Aktaion [Actaeon] sees her bathing after a hunt and the sighting costs him his life.

Having said that, nonetheless, more than a millennium after Aktaion is supposed to have died and after his uncle Dionysos crossed with his armies from Arabia into Africa, the Battle of Marathon took place, with Athens on the defensive against the Persian Empire. This was 490 BC, just a few years before Herodotus’ approximate birth-date.

It was held that the Athenians first began to worship the god Pan after this battle because he showed up to assist them on the battlefield, becoming a centrepiece of their victory. Before the battle, Pan had stopped the Athenian courier Pheidippides on his journey between Athens and Sparta to send him with a message to Athens complaining that they did not offer him honours.

When Herodotus was still a kid, the poet Simonides had already composed a lyric celebrating Pan’s participation in the battle against the Persians. Eventually Herodotus himself records the story, and, after the advent of the Christian era, Pausanias, Nonnus and Suidas also make mention of it. Nonnus claims that Pan’s assistance to the Athenians was prophesied by Zeus during Dionysos’ war against the Indians several centuries beforehand. Aside from Pan, Athena showed up with Herakles and even the long-dead Athenian king Theseus to help out at Marathon.

There were a few superhuman Olympic athletes, all of whom lived in Herodotus’ time. Two of them were the sons of gods; one of them saved a young woman from a ghost which was worshipped in Italy; one was a mad giant with fists harder than stone; and another one was a werewolf. All of them happened to be boxers and nearly all had statues officially commemorating their participation in the Olympic Games.

For more on them, see the next section Superhuman Athletes.

Herodotus writes of an entire nation of werewolves in Ukraine and a race of dog-headed people in other parts of the world, among a plethora of other weird and wonderful beasts, most of whom dwell well outside the bounds of Greek territories.

Herodotus, Pausanias and Plutarch, who render reports of such supernatural individuals, nations and animals, are each quite skeptical of these accounts, the first of these writers especially so. But other Greek and Roman writers like Ktesias (a contemporary of Herodotus) and Pliny the Elder (who lived much later and who cites Ktesias extensively) seem to possess a more open-hearted interest therein.

Ktesias supplies a considerable amount of detail about the populations and lifestyles of the dog-headed people mentioned by Herodotus and before him by Hesiod. St Christopher, the patron of travellers, is supposed to have been one of these part-canine creatures. He was also gigantic in stature, as these monsters were conceptualised during the time of the Roman Empire.

For more detail on encounters with Centaurs, Satyrs (and fauns too), mermen, sea nymphs and winged horses (with giant horns like those of a bull, as distinct from the My Little Pony winged unicorn [“Pegacorn”] variety), see my 3options Answer.

Superhuman Athletes

In the 6th book of his Description of Greece, Pausanias narrates the stories of numerous Greek athletes, many of whom lived in the generation of Herodotus. Among these contemporaries of Herodotus, four renowned boxers stand out as relevant to the present purposes.

Euthymos [Euthymus], the Ghostbuster

It was something like 700 years prior to the writing of Herodotus' Historiai, when

_{Odysseus, so they say, in his wanderings after the capture of Troy was carried down by gales to various cities of Italy and Sicily, and among them he came with his ships to Temessa. Here one of his sailors got drunk and violated a maiden, for which offence he was stoned to death by the natives.}

_{Now Odysseus, it is said, cared nothing about his loss and sailed away. But the ghost of the stoned man never ceased killing without distinction the people of Temessa, attacking both old and young, until, when the inhabitants had resolved to flee from Italy for good, the Pythian priestess forbade them to leave Temessa, and ordered them to propitiate the Hero, setting him a sanctuary apart and building a temple, and to give him every year as wife the loveliest maiden in Temessa. So they performed the commands of the god and suffered no more terrors from the ghost.}

Pausanias had also seen a piece of artwork depicting the Hero ghost as "exceedingly dreadful in all his appearance," wearing a wolf-skin and bearing the name Lykas.

In Book 6 of his Geographika, Strabo tells us that the man who became Lykas had been called Polites and "was treacherously slain by the barbarians" in Italy. Centuries afterwards, now in Herodotus' lifetime, one of the local Italians, an Epizephyrian Locrian by the name of Astykles [Astycles], had a son named Euthymos. Pausanias tells us that, according to these Locrians, Euthymos' real father was actually their local river the Kaikinos [Caecinus], "which divides Locris from the land of Rhegium".

In the decade following the Battle of Marathon, Euthymos competed in two instances of the Olympic Games and then returned home to Italy. He "happened to come to Temessa just at the time when" Lykas "was being propitiated in the usual way," of having a maiden sacrificed to him. When Euthymos found out what was going on, he had a strong desire to enter Lykas's temple and to have a look at the ghost's bride.

When he saw her he first felt pity and afterwards love for her. The girl swore to marry him if he saved her, and so Euthymos with his armour on awaited the onslaught of the ghost. He won the fight, and the Hero was driven out of the land and disappeared, sinking into the depth of the sea. Euthymos had a distinguished wedding, and the inhabitants were freed from the ghost for good.

Classic damsel-in-distress story motif, complete with a happily-ever-after ending which seems to be in place to explain a practice of Euthymos hero-worship; as Pausanias concludes:

I heard another story also about Euthymos, how that he reached extreme old age, and escaping again from death departed from among men in another way.

Theagenes, Born Divine

On both of the occasions in the 480s BC that Euthymos competed in the Olympic Games, a man from the island of Thasos contended against the Locrian hero. This man, Theagenes, was the son of Timosthenes, who was a priest of Herakles Thasios (the Thasian Herakles), although as with Euthymos, Theagenes' real father was said to have been a god.

Herakles had appeared to Theagenes' mother disguised as Timosthenes, and thus Theagenes was conceived. Theagenes can be translated a few different ways, all meaning something like "Divinely Born" or "Born of the Gods." An identical occurrence had led to Herakles' own conception centuries earlier, when Zeus appeared to a princess named Alkmene in the form of her fiancé Amphitryon, and thus impregnated her. A few other feats performed by Herakles in his mortal life are similarly meant as an imitation of Zeus's actions.

When Theagenes was nine years old he got into trouble with his fellow Thasians for carrying home with him the bronze statue of a god from the marketplace. He was punished with the task of putting the statue back in its place, which feat he managed with ease. In adulthood Theagenes landed into some intrigue in his athletic competitions against Euthymos and ended up having to pay a hefty fine. That story can be found in the Description of Greece 6.6.5-6.

Theagenes became victorious in competitions at the Olympic, Pythian, Nemean and Isthmian Games, gaining great fame for himself and his homeland of Thasos, as he won up to 1400 crowns. Apart from boxing he also participated in the pankration [pancratium], the latter of which is essentially an ancient form of mixed martial arts.

After he died, a statue of him fell upon and instantly killed one of his enemies on Thasos Island, for which reason the statue was exiled for murder and, in a circuitous way, this eventually led to Theagenes becoming a deity of healing. The details on that are in the Description of Greece 6.11.6-8. The last section of that chapter reads:

There are many other places that I know of, both among Greeks and among barbarians, where images of Theagenes have been set up, who cures diseases and receives honors from the natives. The statue of Theagenes is in the Altis [at Olympia], being the work of Glaukias of Aigina.

Kleomedes [Cleomedes], the Crazy Giant

In one of the years that both Theagenes and Euthymos participated in the Olympic Games there was another boxer present, called Kleomedes, from the island of Astypalaia [Astypalaea]. Calculating this by notes from the Description of Greece translation by William Henry Samuel Jones and Henry Ardene Omerod, the year would seem to be 487 BC.

Plutarch describes Kleomedes as a huge man. The 1894 translation of Plutarch's Life of Romulus by Aubrey Stewart and George Long quotes Plutarch as saying that this athlete "was a man of unusual size and strength, but stupid and half-crazy". Bernadotte Perrin's 1914 translation says that he "was of gigantic strength and stature, of uncontrolled temper, and like a mad man".

Pausanias says that Kleomedes killed Ikkos of Epidauros during a boxing-match at the games. Accused of foul play, Kleomedes lost the victory prize and returned home mad with grief. Continuing along with Plutarch's account: now back on Astypalaia, while at a boys' school, by the blow of his fist, Kleomedes broke into two pieces the pillar supporting the roof of the school, bringing down the house upon the children inside and thus killing these minors, whom Pausanias numbers as sixty boys.

According to the Description of Greece, Kleomedes' fellow citizens pursued him, pelting him with stones so that he took refuge in a temple of Athena wherein there was a big chest with a lid. Kleomedes jumped into the chest and pulled the lid over it. Says Plutarch, he was able to hold onto it so tightly "that many men with their united strength could not pull it up". Pausanias' conclusion to the story goes like so:

^{At last, however, they broke open the boards of the chest, but found no Kleomedes, either alive or dead. So they sent envoys to Delphi to ask what had happened to Kleomedes. The response given by the Pythian priestess was, they say, as follows:}

^{Last of heroes is Kleomedes of Astypalaia;
Honour him with sacrifices as being no longer a mortal.}

^{So from this time have the Astypalaians paid honors to Kleomedes as to a hero.}

Damarkhos [Damarchus], the Shape-Shifter

It was said that in the secret cult of Zeus Lykaios [Lycaeus], the god of Mt Lykaion in Arkadia [Arcadia], every nine years a group of the cult's votaries would sacrifice a boy and mix his entrails with the intestines of some animals. Then they would separate the part of the mixture which had human flesh in it and cast lots to see who would receive that portion.

Whoever did so would then have to eat the meat and then take off his clothes, swim across the nearby river and go into the wilderness where he instantly transformed into a wolf. He would remain in this state for eight years starting at that point and if by the ninth year he had not eaten any human flesh then he would become a man again, swim across the stream and rejoin the community on the other side. The cycle would then begin again.

In 400 BC there was a boxer from Parrhasia in Arkadia who participated in the Olympic Games. His name was Damarkhos, son of Dinytas, regarding whom there was a story that he had transformed into a wolf for nine years, on account of his participation in the worship of Zeus Lykaios, before becoming a man again.

Pausanias says that he cannot believe the tale, regarding which not even the Arkadians themselves have any record. Surely, thinks Pausanias, the inscription on the statue of Damarkhos would have mentioned this detail about his life rather than just his name and birthplace.

_{However, the transformation was recorded by a Greek writer, Scopas, in his history of Olympic victors, who called the boxer Demaenatus {Demainatos}, and said that his change of shape was caused by his partaking of the inward parts of a boy slain in the Arcadian sacrifice to Lycaean Zeus. Scopas also spoke of the restoration of the boxer to the human form in the tenth year, and mentioned that his victory in boxing at Olympia was subsequent to his experiences as a wolf. See Pliny, Nat. Hist. viii. 82; Augustine, De civitate Dei xviii. 17.}

_{Sir James George Frazer's translation of and commentary on Apollodorus' Library, n. 3 on Book 3 Chapter 8}

Just a few decades before Damarkhos, Herodotus writes of a similar werewolf story, set in what is now Ukraine and Belarus. He is unable to take the information seriously, as he reports in his Historiai 4.105:

_{The Neuroi [Neuri] follow Skythian [Scythian] customs... It may be that these people are wizards; for the Skythians [Scythians], and the Greeks settled in Skythia [Scythia], say that once a year every one of the Neuroi becomes a wolf for a few days and changes back again to his former shape. Those who tell this tale do not convince me; but they tell it nonetheless, and swear to its truth.}

Semi-Divine Royalty

In 356 BC, roughly seventy years after the death of Herodotus, and less than fifty years after Damarkhos' participation in the Olympics, a son was born to King Philippos [Philip] II of Makedonia [Macedonia]. The child would grow up to succeed his father on the throne, becoming Alexandros III of Makedonia, but much better known in English as Alexander the Great. In much more dramatic accounts than those of Herakles and Theagenes, Alexandros was reputed to be the child not of Philippos but of a god.

The Greeks believed that the Egyptian god Amūn, whom they called Ammon and who was worshipped in Libya as well, was the same as their Zeus, and he was often even referred to as Zeus Ammon. This deity had an oracle in Libya. When Alexandros came to this place the god is supposed to have spoken through the oracle in order to acknowledge him as his own son. Shortly after Alexandros' death there were statues made of him sporting a pair of horns because his father Ammon had the head of a ram.

In the more Greek versions of this tradition, the king's divine father appears as a dragon (drákōn, in this case more of a huge monster snake rather than the winged, fire-breathing variety from modern Western fantasy) which, it seems, is supposed to be Zeus. This is connected to the Orphic story in which a primordial incarnation of the god Dionysos [Dionysus] is sired by Zeus in the form of a drakon.

_{[O]n his father's side he [Alexandros] was a descendant of Herakles through Karanos [the first historical king in the lineage, reigned c. 800s BC], and on his mother's side a descendant of Aiakos [Aeacus; son of Zeus] through Neoptolemos [son of Achilles and grandson of Aiakos]; this is accepted without any question.}

_{And we are told that Philippos, after being initiated into the mysteries of Samothrace at the same time with Olympias, he himself being still a youth and she an orphan child, fell in love with her and betrothed himself to her at once with the consent of her brother Arymbas.}

_{Well, then, the night before that on which the marriage was consummated, the bride dreamed that there was a peal of thunder and that a thunderbolt fell upon her womb, and that thereby much fire was kindled, which broke into flames that travelled all about, and then was extinguished...}

_{Moreover, a serpent was once seen lying stretched out by the side of Olympias as she slept, and we are told that this, more than anything else, dulled the ardour of Philippos' attentions to his wife, so that he no longer came often to sleep by her side, either because he feared that some spells and enchantments might be practised upon him by her, or because he shrank from her embraces in the conviction that she was the partner of a higher power.}

_{However, after his vision, as we are told, Philippos sent Khairon of Megalopolis to Delphi, by whom an oracle was brought to him from Apollon [Apollo], who bade him sacrifice to Ammon and hold that god in greatest reverence, but told him he was to lose that one of his eyes which he had applied to the hinge-gap in the door when he espied the god in the form of a serpent, sharing the couch of his wife.}

_{Moreover, Olympias, as Eratosthenes says, when she sent Alexandros forth upon his great expedition, told him, and him alone, the secret of his begetting, and bade him have purposes worthy of his birth. Others, on the contrary, say that she repudiated the idea, and said: "Alexandros must cease slandering me to Hera [Zeus's wife]."}

Plutarch, Life of Alexandros 1.2-3

In Drakon: Dragon Myth and Serpent Cult in the Greek and Roman Worlds, Daniel Ogden says: "By the mid 4th century AD the tradition had become so well entrenched that Alexander could be addressed by the epithet drakontiadēs". The epithet denotes "son of a dragon," incidentally the same thing that Dracula means.

In much admiration of Alexandros, similar traditions became part of Roman culture centuries after his time, most notably with the first Roman Emperor Augustus Caesar having been said to be the son of Apollo, who had impregnated Augustus' mother Atia in the form of a dragon.

ESCAPE FROM THE WORLD

Next after the remarks made by Pausanias on the issue (see here, under Option #2), the most explicit example of such a thing as an explanation for the absence of visible deities seems to come from a Roman rendition of something said by Hesiod in his poem Works and Days, wherein he provides a breakdown of human history in five stages or ages. The ages are the Golden; the Silver; the Bronze; the Age of Heroes (which was brought to an end by the Trojan War); and the Iron Age, in the last of which Hesiod lived and which apparently continues in increasingly worsening conditions today.

Works and Days forecasted that later on in the Age of Iron the goddesses Aidos and Nemesis were going to shroud "their sweet forms in pale mantles" and escape from the world and its corruption to go up Mt Olympos [Olympus], forsaking "humankind to join the company of the deathless gods, and leaving nothing but bitter sorrows for mortal men. And there will be no help against evil."

About half a millennium after Hesiod, Aratus writes a poem entitled Phainomena in which he explains the origin of the constellation Parthenon, the "Maiden," which is called Virgo in Latin. There was a Titan goddess named Astraia [Astraea], the "Starry" One, whom mortals refer to as Dike, "Justice." She dwelt freely among the men of the Golden Age in a time of peace when she would gather together the elders of towns in order to share her wisdom with them.

Then came the Silver Age, which made her reticent about hanging out with humans, although she did continue to meet with them. Finally when the people of the Bronze Age began to use bronze for daggers to rob wayfarers and when they started "to eat of the flesh of the ploughing-ox", she couldn't take it anymore and fled into the sky, where she is now visible as the "Virgin" constellation.

A couple of centuries after Aratus, the Roman poet Ovid takes this up in his poem the Metamorphoses, wherein he seems to combine Hesiod and Aratus by saying that it was during the Iron Age, perhaps at its onset, that Pudor, Veritas and Fides (Modesty, Truth and Loyalty, whom the Greeks would have called Aidos, Aletheia and Pistis respectively) were the first virtues or powers to flee from the world. The situation got gradually but dramatically worse before finally Pietas (Piety) was vanquished and Astraea became, according to A.S. Kline's translation, the

last of all the immortals to depart

and "herself abandoned the blood-drenched earth."

Ovid's version of the decline of virtue in human society reads a lot more strongly as corresponding to a steady decline in the number of the gods (at least the peace-loving ones) who dwell on Earth. Ovid's connection of this with the time of the Titans (Astraea herself being a Titan) is made even stronger since in his version of Man's Ages, the Golden Age was the best because it was governed by Jupiter's father Saturnus (the Roman equivalent of Zeus' father Kronos).

That age came to an end because Saturnus was ousted by Jupiter, in effect making Saturnus's exile the beginning of the gods' flight from the world of mortals. (The sequel to this story appears in Works and Days, as well as in the writing of Pindar, both of which say that Zeus was eventually reconciled to Kronos, whom he installed as king over the Islands of the Blessed, where humans who were deemed worthy of a pleasant postmortem existence would dwell after their deaths.)