Yeah I suppose. Religion is often not really chosen, but kind of forced onto a person early in life. I suppose the same is likely true for neo Nazis though.
Of course they do. It's impossible to know whether or not God exists based on physical evidence; that's why religion is based on faith and not scientific research. People choose their religious beliefs, or (much more commonly) have them chosen for them. That's faith.
Yes, I think they should, if they could somehow be confident that their guests would dine in peace and not make trouble. The fact that such confidence could be hard to come by shows that this situation is not really analogous to the wedding photographer situation. You'd have to flip it around to get a better analogy: should a neo-Nazi restaurant be compelled to serve Jews? Clearly yes.