First you have to finish putting up the sheetrock. The tapered edge should not be around the door frame. The sheetrock should be closer to the jamb then where it is now, it needed to be about 1/2" from the jamb.
Once the sheetrocks up, I would finish it and prime and paint it so you not stuck getting compound and paint on the trim.
The frame as you call it (it's really called the Jamb) just needs a jamb extention so it will sit even with the drywall. Just measure the distance from the jamb to the sheetrock face and add a piece of wood that thickness around the face of the jamb. keep it back from the edge by 3/16 to 3/8.
(look at how the other windows and doors in the home were done and match it) Make sure to preprime and paint the piece so paint does not get on the jamb.
Now your ready for the casing (the trim around the frame of the door).
If you know what type floorings going in you can cut the casing on the sides shorter to allow the new flooring to slide in under it, instead of having to go back and under cut it later.