The entire semiconducter is doped and has the same concentration throughout.  Basically, holes leave the p region and electrons leave the n region in the vicinity of the junction.  So, this is where the depletion region is, and it is made up of ionized bound atoms (the region is depleted of carriers).  The rest of the semiconductor has dopants, but they aren't ionized.

The depletion region is always at the junction, regardless of the doping profile.

You'll have to read up on diffusion and drift.  Diffusion occurs because of difference in concentration.  The p side has lots of holes, so holes go to the n side, and vice versa.  This depleted region sets up a field that counteracts the diffusion current.  At equilibrium, the diffusion current equals the drift current and the depletion region is held at a constant size.

If you disconnected a pn-junction then yes I think the carriers would redistribute equally.  However, I'm not certain each piece of material would be neutral.  The atoms might still be ionized if there was no way for the electrons to get back to the n-material, for example.

Cheers,
Marc