In case someone comes across this in a future search, here is what I did on a similar project:
First, I used an oscillating saw with a plunge bit to cut the nails—-much easier than a sawzall. I used little magnets to sweep along the face of the stucco mould and locate them, then I plunged in from the side to cut them. I cut the paint with a utility blade along the seam where it joined the frame and tapped the mounding free with a hammer & chisel (aka screwdriver).
Then I cut the door frame nails with the plunge bit from the front and back after locating them the same way. Since my door also had zero gap between the frame and the studs, I had to do some persuading with a hammer to move the stud where I wanted it, plus a slight trimming of the bottom plate with a hand saw. My header was also nailed out of level and I had to tune up the corner with an actual chisel.
Now, my stucco molding was 1” thick and they apparently do not make that any more, so it was impossible to replace (would have been easier to paint and reuse it but it was in rough shape). I had to go with the 3/4” stuff. I also went with a door frame that was ~1/4” wider to compensate (old was 4-1/4”, new was 4-9/16”). The shape of the channel turned out to be different as well, so I simply cut off the exposed lip of the stucco with an angle grinder with a diamond wheel. I also had to rip off the lip of the stucco moulding on my table saw.
I had to cut the top of the stucco about an inch higher to match the new, standard sized door frame I bought from the big box store. I used a grout blade for my oscillating saw to do the corners. To finish I used OSI Quad caulking between the stucco mounding and stucco and it turned out great, no need to repaint anything because I cut a nice clean edge.
As you can tell, I ran into every single nightmare possible! But I could see all of them coming, had the right plan and more importantly the right tools. I wanted to share my experience in case someone else is looking to do the same job. Good luck!