I heard from somewhere that gods maintain their power from the people worshiping them. The more people pray, the stronger they get. I also heard that If their home/environment is destroyed (eg. Posideon's waters being polluted) then they will lose some of their power.
What I would like to know is...
Is this true?
Can they be killed if no one cares about them?
I don't know about how true it is -- really, how can we? But it is an ancient idea, first put forth by Plutarch, who posited that Pan had died because people no longer believed him to be more than a story (In "The Obsolescence of Oracles"):
The power of the spirit does not affect all persons nor the same persons always in the same way, but it only supplies an enkindling and an inception, as has been said for them that are in a proper state to be affected and to undergo the change. The power comes from the gods and demigods, but, for all that, it is not unfailing nor imperishable nor ageless, lasting into that infinite time by which all things between earth and moon become wearied out, according to our reasoning. And there are some who assert that the things above the moon also do not abide, but give out as they confront the everlasting and infinite, and undergo continual transmutations and rebirths."(section 51)
(That idea becomes a subplot in The Percy Jackson series).
While I don't know of any Greek or Roman sources that discuss this topic in complete seriousness, it does appear in the comedy of Aristophanes (fifth century BCE)—which indicates that the idea existed in the public consciousness to some extent.
In the Birds (aka Ornithes in Greek, Avēs in Latin), an army of birds and transformed humans bands together to overthrow the Olympians and take their place. Their plan relies on building a floating city called Nephelococcygia ("Cloudcuckooland"), from which they can blockade Olympus, preventing human offerings from reaching the gods.
This scheme does in fact work, and the gods promptly begin to starve for lack of sacrifices. It's implied that they're literally starving to death (the whole thing is a reference to the Athenian blockade of Mēlos a few years earlier), and in the end Zeus is willing to give up pretty much anything (including a lot of his godly power) in exchange for ending the blockade.
Now, Aristophanes is a comedian first and foremost, and everything he writes about mythology should be taken with many grains of salt. Nevertheless, this does show that Greek authors were thinking about this idea well before Christianity took over.
Carl Jung said “The gods have become diseases; Zeus no longer rules Olympus but rather the solar plexus, and produces curious specimens for the doctor's consulting room, or disorders the brains of politicians and journalists who unwittingly let loose psychic epidemics on the world”
God’s can’t be killed as such and mortal humans can’t increase or decrease a god’s own personal, innate power with physical acts. When forgotten, they merely fall below the threshold of human consciousness and act on humanity via the unconscious as per Jung’s quote above. Your question presupposes a very literal and physical existence for the gods whereas mythology is really allegorical and their true nature is more psychological (or spiritual) - although no less real for being so.
Also, the story of Pan’s death may well have arisen from a mistranslation of the Adonis ritual where the celebrants re-enact his life and death and shout out at some point, Adonis is dead. This was erroneously translated as Great Pan is dead which seemed to chime with a general belief at the time that the pagan religions were in terminal decline.
Yes, it is true. Cannot give you scientific proof but that's what I understood on my spiritual journey :-) And it's not only truth about gods, but also about places, statues... for example. If you come to an Indian temple where people worship with pure heart and pray every day... and you are sensitive enough, you will feel how holy the place is, you will "feel the god" there. On the other side, if you buy a statue of a god, bring it at home, and if it will be for you just a statue from a vacation, it will not have that power that you feel in the temple. Whereas if you would pray to that statue every day, that power would be there. I am now at a point where I can feel these things - if I enter some space, it is very different if someone takes care of it spiritually or not. It's like cleaning, but on different levels :-)
There's also another New Age metaphysical teaching, represented e.g. by Teal Swan. And it perfectly fits to mythology as well. According to this teaching, there are a lot of beings that are interested to interfere with humans on Earth, since this is probably the most interesting experiment in the Universe :-D Some of those beings are here to help, some of them can be parasitic (like demons). People can have a lot of conscious and unconscious contracts with these beings and their power is basically proportional to the attention (conscious or unconscious) that they get from people.
According to the second part of your question, I think they cannot be killed. Anyone who would remind and remember them would again put them to life back again.
