1. This is referred to as a "lotus birth". Lots of threads on MDC about it. Do a little searching.
2. I have no idea. Usually they'll let you keep the placenta if you tell them in advance, but I'd be quite surprised if you found a hospital that would let you keep it attached. Never hurts to ask.
3. You'll probably get laughed at (or at least looked at like you're crazy). But who cares? The world is a "don't ask, don't get" place. Again, never hurts to ask. Still, since you're having a hospital c/s, I suggest picking your battles to make sure you get the things that are really important to you. Chances are they will be flexible on some things and not others. If this is one of the things that is VERY important, make sure they know that. If not, I'd ask and drop it if they say no way.
Mom to James ( 5/2006), Claire ( 6/2008), furry kitties Calvin and Bob, and wife to Dennis.