Honestly, as long as the person isn't extreme (like a member of the "God Hates Fags" church) I would not mind their beliefs. I'm a very open-minded dude and as long as they aren't just complete idiots (like strangeluv's number 3) I'd be okay with whatever they think.
Here, I'll answer yours too.
If you're a registered republican, would you go out with a registered democrat in search of a real relationship?
Since I am a registered republican (how did that happen??) but subscribe to very democratic views, I'd be okay with that. PS I changed the question to be more applicable to me. I'd also attempt a serious relationship with a registered republican. Honestly political views do not matter
If you're an atheist, would you go out with someone with religious beliefs?
Or maybe you'd go with a buddhist, but not with a christian? Or similar variations.
religion doesn't matter unless they're trying to convert me to their beliefs or they have really messed up beliefs (sacrifice manny baby to serpent god Xthiliastrxma)
If you're pro-choice, would you go in a serious date with a pro-life person?
No. Too many things would go wrong and honestly a lot of pro-life people I've met have a lot of other personality flaws and aren't that great people to be around oh well!!
You believe in capitalism, the other is a commie. Would you proceed?
If they were just casually "well it would be better this way" then I'd be fine with it. If they were all hardcore going to rallies and yelling at me for supporting capitalistic pigs or whatever then wow, no thanks. Honestly though, I'm a bit of a commie myself (according to that political compass I was a lotta bit a commie!!)