Authoritarians are very possibly not welcome. Non-authoritarian members of the religious right, cultural conservatives, etc. are very likely welcome. (And of course any variety of libertarian.)
I think that depends, and the issue is more complicated than you make out. To play devil's advocate, consider the conflict between freedom of association (or religious freedom) and anti-discrimination laws. You could argue that valuing either is a pro-liberty stance. The notion of liberty is kind of difficult to pin down when you're talking about interactions between free individuals.
There are some people out there that are culturally conservative personally, but don't believe in forcing their beliefs on other via laws. Not many, but they do exist. And they generally vote libertarian.
