To speak to the first question: we are usually led a bit astray by the term "myth," by which we tend to mean a story somewhat akin to a fable that is (to us) obviously not true. To the Greeks, "myth" just mean "story" or "plot." Their religion was in part made up of a lot of stories, but so is every religion. That doesn't mean they didn't believe them or that they didn't have a flexible understanding of their literal vs. figurative truth (i.e. you don't have to believe everything in Bible to believe in God, etc.). Generally, Greeks were fairly religious well into Roman rule.
Also, a fifth century Greek's understanding of the concept "history" is hard to pin down. While someone like Herodotus (and especially Thucydides) tried hard to distinguish fact from fiction, they didn't really engage in some of the central methodological practices that we do (periodization, scrutiny of sources, primary sources), so although we can talk about ancient historians as historians, it is useful to keep in mind that "history" had much different connotations to them than it does to us.
For example, it would be fine back then to say "The Trojan War definitely happened, although I have no evidence, and it happened some time a long time ago."
I also want to second @Gracie K's point: till recently, it was standard to incorporate the bible into historical works. And still, there are plenty of popular history writers who take religious content as fact.