The med school absolutely would care - these positions can be extremely competitive, so you want a solid post-doc with tons of opportunities for authorship, and the potential to establish your own research line and get extramural funding. That doesn't mean anything besides Harvard = failure, but I imagine a good one would open doors for you that wouldn't be there otherwise.
Whether you could charge more in private practice probably depends on a huge number of factors - I imagine your post-doc is pretty far down that list. With insurance companies....seriously doubt it would make ANY difference, unless you're gaining competence in a specialty you otherwise wouldn't be qualified in.
I'm nowhere near the post-doc stage yet, but I can't imagine taking anything below the NIH payline (37k or thereabouts right now for the first year) unless something drastic changes. Plenty of places pay more, and from everything I have seen and heard, postdoc positions are plentiful and not overly difficult to find.