Hello Tolulah,
Yes, a clinical center is basically a place where you can do all your cores. This is ideal for married couples, or people that just don't want to be constantly moving around. if you sign up and get a clinical center, you are required to stay there for at least one year (to complete all the cores). then for your fourth year, you can go do electives whereever you want...so,
1. yes, if you begin at a clinical center, you must finish there (one year later)
2. see above
3. well, i'm still in basic sciences in St. Vincent, but finding out the diffs between rotations at individual hospitals is hard without speaking to the students who have done them! some just have better teaching programs than others and it depends on what field you want to go into!

hope this helped even a little bit!!