Correct. You could trim the door by butting the trim up against the door frame instead of on top of the frame. If you get thicker trim no one will every notice (except you).

I would install a piece of trim to cover the door latch / lock holes. If installing trim around the door make sure the trim is wide enough to support the trim. Paint white to match door trim. Install the drywall first and then the trim will cover the edge of the drywall.

This would probably work so that no one else notices. You will notice it everytime you look at it - if you can live with that then I would go with your idea.