I have wondered what a cab would look like if just the back was
faceted instead of being flat.
Well, of course, then it would no longer be a cab... In order for
the back to be faceted, it would stick out like, well, a faceted
stone instead of lying flat. This is called "buff top" and is done
sometimes. But it is "neither fish nor fowl", and IMO, stones
generally look best either cabbed or faceted, not in between.
Incidentally, the o[pposite is done too-- faceted only on top. This
is called rose cut, and can be very nice, especially on stones that
are too dark or included for light to bounce through them, like
marcasite or some garnets.
A rounded back is pretty common, and again IMO it does not
accomplish much except to make the stone harder to set. It also
makes the stone into a magnifying glass (if it is transparent), so
it becomes very important what the metal under the center of it