I can relate to somebody wanting to have something to believe in.
I have two concerns with my work: having good things to act, and getting paid. In that order. Although if you're not getting paid well, that order can change. But that's what I'm concerned about. Good scenes. Decent money.
I don't think I could play a character that I couldn't relate to somehow. I'm not unfamiliar with frustration, anger, shame, helplessness and a load of other emotions that make up our psycho-soup. I try to focus on that frustration, that sense of unfairness, and multiply it.
I so rarely turned down a role, that I can't say I have any regrets in that regard. There were many roles that I would rather not have done, but having a home and family requires that we sometimes do things we would rather not. I have done a few roles that I've never watched, and if I happen to be flicking through channels and one pops up, I quickly move on. It's hard enough to sustain some self confidence without being reminded of things we'd rather not revisit but, in the end, it comes with the territory.