Did the Moirai / 3 Fates take precedence over the Gods responsible for love and marriage? What relationship, if any, does Greek canon state about the link between Hera, Eros and Moirai?
It is interesting that in Chinese lore, a red thread was said to connect lovers. Note that the actual name is the "Red String of Fate".
The Chinese tradition does not have a direct analogue to the Moirai so far as I know, and there tends to be more emphasis on "fortune" as in bad/good, presided over by the god Caishen. However it is telling that one of the "Four Great Folktales" of China is the Weaver Girl and the Cowherd, which is a love story, and the Red Thread of Destiny is understood in a romantic context, where Yue Lao serves the function of a god of fate.
Weaving and fate tends to be intermingled in the West as well, which is why the Greek fates are female (weaving was understood as a source of feminine power), and a human life was understood to be a string that Atropos would cut when a person's allotted length of time was up.
It was said that Hymen, the Greek god of marriage, could bless or curse a union with his presence or absence at the ceremony. But is Hymen influencing fate, or merely acting as fate's agent?
As you astutely point out, Eros, does seem to be able to influence destiny with his arrows. He makes Psyche fall in love to punish her.
According to Hesiod, Eros, who is sometimes referred to as "both the oldest and the youngest of the gods", comes into existence before the Moirai.
Aristotle, quoting Hesiod, notes the implication that Love (attraction) must be the prime force of creation:
And Hesiod says, “ First of all things was Chaos made, and then/Broad-bosomed Earth . . ./And Love, the foremost of immortal beings,” thus implying that there must be in the world some cause to move things and combine them.
"Hesiod... assumed Love or Desire as a first principle in things."
It becomes a philosophical question at this point: Was it Psyche's fate all along to be so punished? Is Eros altering destiny or merely acting as its agent? If Eros is the prime force behind all creation, is not fate a function of Eros?
Aeschylus, Eumenides 971 ff :
"[The Eumenides bless the Athenians with good fortune :] ‘I forbid deadly and untimely fate for men; grant to lovely maidens life with a husband, you that have the rightful power; you, divine Moirai (Moirae, Fates), our sisters by one mother, divinities who distribute justly, who have a share in every home, and whose righteous visitations press heavily at every season, most honored everywhere among the gods!’"
They have the power to make women fall for men,
Aristophanes, Birds 1720 ff (trans. O'Neill) (Greek comedy C5th to 4th B.C.) :
"Twas in the midst of such [wedding] festivities that the Moirai (Moirae, Fates) formerly united Hera Olympia to the King [Zeus] who governs the gods from the summit of his inaccessible throne."
and arrange godly marriage.
As @DukeZhou says, Eros was one of the first forces of Love. And probably the strongest.
But, according to Aeschylus, they have the power to give a maiden a husband.
What I view from this, is that Eros is more of the falling in love sorta deal, while the fates are about finding a spouse. Hymen is the wedding type of guy, rather an agent of fate. Hera... I honestly have no idea.
First we have to understand what kind of love there is. I will assume we talk about eros love. Eros love is divided in the aerial love of feelings and the earth love of sexuality. Eros acts as a trigger. This is why Eros is one of the primal gods. Eros always tries to create and unite. Eros triggers the unknown and the will to find or create something. So he is not responsible for what it follows. Now let's understand moirai. Moirai are three United goddess that follow a man's path. Klotho. Klotho is the one that will decide if the eros will create a man between a man and a woman. Lahesis. Lahesis is the one that will decide what will happen to a man came in life whether is good or bad. Atropos. Atropos is the one will decide when a man's life will end. So for sure Lahesis has an important role on what will happen in a man's life concerning his erotic life after eros joined them. They may fight or even divorce or live happy or be successful couple. But in no way Lahesis can ensure that your happiness with your lover is also the others happiness. Because Lahesis and all moirai act independently to all.