Transition between casing and baseboard

Your plan to fix the gap in the center of the door sounds fine. I might suggest using some shingle shims instead of flat stock however. If you want to just remove the interior trim only, that is fine also. Using the shingle shims, place the thickest end in first and by trial and error, adjust the opposing shim to a size that will snug in nicely and then by gently tapping it in next to the first shim, remove the excess gap. Put a small finish nail through the jamb and shims to prevent them from moving. Drilling a pilot hole will assure not splitting the shims with the nail in inserted. Now just trim off the excess and reinstall your trim. Job done! Good Luck

I can think of two options using the prehung doors available in HI stores:

You can pull the door apart (they come in two pieces for the installation) and cut about 3/4" off of the inset part of the door. Since this side goes into the channel on the other half of the door, you'll never see it if the cut isn't pretty. I believe there's at least 1/4" of play in that channel, but measure first.

Second option is to get some strips of 1/2" wood to install behind the trim around the door. Hang the door as you normally would, but leave it 1/2" out from the walls. Then, before nailing the trim into the wall, place the 1/2" strips of wood behind the trim around the doorway.

The first option is a bit tricky since the door is very weak when disassembled and the trim makes the corners difficult to cut around. The second option comes in handy when you realize the problem after you've already installed the door (we all forget to measure first sometimes), but I don't think it looks as good to have trim sticking out like that.