My ripe (or at least ORANGE) pumpkins seem to be getting mushy stems
right at the vine and then that's that. The first one that turned orange
turned out to be completely hollow inside due to rot although the outside
looked picture perfect. These are not large ones, either, although there
are a few larger ones still on the vine and still okay, stem-wise.

Any idea why they are going mushy at the stem/vine connection?

I've noticed that small pumpkins will sometimes drop off of the vine. I'm
guessing the vine can only support so many pumpkins, and any excess are
dropped. I wonder if that applies to big ones also?