Nice looking garage.
I'm guessing you are going for a clean smooth look in the end if you're taping, mudding, and painting.
With that style openers, the header and inside "casing" is even more visible vs. big ol glop flinger springs.
I wanted mine to look clean when I finished it out. I taped and mudded everything up to the framing before painting. I mention that only because I couldn't even remember if I painted the exposed lumber around the doors. I had to go back out and look before posting.

It must not be a huge deal to me because I realized I had only painted the header and framing across the top. The side strips are still bare wood.
I finished my garage twenty years ago. I've redone door rails since that time and I guess it worked out the side framing was not painted. On the other hand, the bare wood does stand out with everything else painted. Your garage is nicer than my setup. It will be more noticeable.
The gap between framing and drywall in one picture looks fairly big for caulking or filling with mud. I've used spray foam instead of backer rod in spots with that type of gap. After the foam sets, I cut it below flush so I can try a smooth patch across the gap. It's certainly not a perfect solution but it serves two purposes. You're doing gap seal and patch at the same time.
I'd definitely agree with the post about patching knot holes, etc. if you paint the framing. In fact, I'd replace the piece in the second picture with a better chunk that has a square edge.
All this work is so much easier to do now instead of later. Definitely a good idea to be proactive on the finishing.