You'll get slightly different answers from different people.
In many cases, the religion has no opinion. Then they make the decision about like you would, I'm guessing. Gut feeling, weigh the pros and cons, go with what's convenient or, given your experience and perceptions of your experience what is most likely to be veridical. (I'll use the word 'veridical', which just means 'true', to mean "coinciding with observed reality" or what mere observed reality would predict.) Monsters under beds, lizard people in government.
Dating is a bit more complex, of course, because then you're also evaluating not just easily tabulated pros and cons but emotions, compatibility, etc. Then religion, being a signal of compatibility in ritual and morality, tends to come into play, along with convenience in avoiding bloodshed when it comes to how to raise the kids.
In many cases, the belief has no consequence. The dinosaurs went extinct before the flood? Whoopee. The US and Britain are the descendants of the lost tribes of Israel? So what? A lot of prophecy falls into this; a lot of it is just there to reconcile other beliefs with what outsiders say in order to avoid controversy or, about as often, to avoid the outsiders from making claims about core doctrines that often involve ritual or moral claims. It's when people start insisting that you believe X or when the claims start hitting more important doctrines that you get real pushback. Evolution's only a big deal for most people because of what follows afterwards, so it gets an easy kludge. Then the criterion used is, "Does it safeguard other beliefs that I cherish?" In a few cases of absolutists these fringe "doctrines" are made into large issues, mostly because they relish fighting as much as their opponents.
To the last three questions.
Different faith systems have different criteria. So Episcopals have the text of the Bible (however that get's interpreted), tradition (both bible interpretation and other stuff), and "the holy spirit" which often means "whatever the hell we think is a good idea right now." Other faiths rely on "the holy spirit" and "inspiration" more or less.
Some are strictly text-based, with tradition or principles guiding how to interpret the text. So believers, faced with a new claim or situation, go to the text. Gut feelings work their way in--"I can't believe the text says that" is followed by a huge "study" to disprove it, with the criteria for disproving getting lower and lower. Or "I need the text to say that"--again, it's motivated thinking. In some cases there are traditional intepretations or appeals to human authority. "He's the apostle of Gawd!"
Some Islams are like the Episcopals. Some like more text-based. Some have the Sunnah, which are traditional stories to fill in cracks and provide context; all rely on argumentation, which you're often free to accept or reject but which pattern into fiqhs or schools of jurisprudence.
Some doctrines are more important than others. My old church would have no problem with gay rights; it would explode with fury at gay marriage. It would see no problem with the difference. It kicked out unmarried people who insisted on having sex, and beyond that didn't care who you wanted to lay. It really insisted on Jesus' "in the beginning" line to dispose of divorce and Paul's "husband of one wife"--taken not just to mean "no polygamy" and "only heterosexual marriages" but also "no divorce." So if you were gay, you wouldn't marry. If you'd been married, you also woudn't marry. In fact, if you had doubts about a relationship, you tended ... not to marry. Otherwise you're stuck married and unhappy or stuck divorced and unhappy. At the same time, if you celebrated your birthday or accepted Xmas presents, meh.
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