This answer depends on the office that you are in. If you are in Chicago or Boston, you may have similar overlap in terms of your experiences with those who come fro NY/SF. This is due to the fact that the nature of work in Chicago / Boston / other mid-large sized cities is on larger, Fortune 500 corporations. If you are coming from McKinsey Stamford, Detroit or Pittsburgh, you certainly won't have the types of exit ops someone from Chicago / Boston / NY / SF people would.
2. How easy is it to switch offices once you start your job full time?
At one of the MBB, I've seen at least 5 coworkers do it near the first year mark. The wwitch depends on how strong a performer you are and second what your reason is (e.g. personal, near family, etc. that stuff is weighed more heavily than just "I want to try a different city")
3. Can you switch offices after being sponsored for business school?
Yes. However, keep in mind, though you are not obligated to go back, you may be encouraged to. There are some plus sides to being in a smaller office: you know more people in the office, if you are a true stand out you could get promoted earlier, etc. As a generalization, the people in NY/SF can be a notch higher, so the competition is less at places like McK Pittsburgh, etc.