Definitely go past the door.

I would go past the door to eliminate any issues with moisture getting too close to the drywall. You don't have to go a whole tile past the door though (I'm afraid that might look strange depending on the orientation of the shower layout in your bathroom).

What you are using for finishing off the exposed edges where the shower tiles end? The reason I ask is because you can always use a Bullnose tile that are typically about 3-4" thick, or use the field tile you have and cut it down (to maybe half?), and finishing it off with a strip called Schluter. They are thin metal profiles that are shower safe that are a good finishing edge alternative for tiles that do not come with a Bullnose piece.

The typical length to extend the tile past the threshold is about 4"-5".

PS- mesh over the breaks in the sheet rock and red tape (or any type of waterproofing membrane) needs to go on before tiling that wall. Hoping the other walls were waterproofed before tiling began.