I'll attempt to get over my
at some of the questions and just comment on the "do I fit in" piece... Think of it in some aspects like a job interview, except more personal. Some employers like it when the candidate is bold and says "Why do you think I would be good for this role?" or "Why should I choose you?". Other employers are put off by that and would rather hear "what do you look for in X" or "this is how I can help you with Y". It's active listening, really. Sometimes it's better to leave the "I" out of it (for part of the time at least) and get more into the "you". You can show your interest that way.
If you want to come off as sincere, ask about the things that really do matter to you. Remember that it's a mutual selection process, so you're trying to find the right group for you as much as they are trying to find the right brother for them. Think about the reasons you want to join and frame questions around those topics. Try to get a feeling for the personality of the group, potential leadership opportunities, personality traits/qualities they look for, hobbies, interests, time commitment, community involvement, campus involvement, etc. You don't have to grill them, but when you walk away from the week you should have a sense of who they are, what they stand for, and how you might fit in.
Those are just my thoughts for the moment. And honestly, if a group asked me some of those questions I'd probably engage in some polite sarcastic banter and never come back. But to each his own.