1. Ask a professor who knows you and who you trust. Alternatively, apply to as many as you can and figure it out at the interviews.
2. It depends on how well you know your adviser and what he thinks of you, since he is the one who's going to write your recommendation.
3. I think most applications ask you for research interests, but not for a research proposal. It can be really vague. Mine was.
4. There have to be some faculty members with overlapping research interests somewhere at the institution. Otherwise, you won't find a lab there you want to join, and none of the labs will want to take you. The people you talk to don't necessarily have to be the people with overlapping research interests.