I'm removing an exterior door and window and have some questions.

I know the easy way is to just frame in and sheet over the window/door opening. But that also lends to a large amount of thermal bridging where all the studs/headers etc are. Part of the reason for removing them is the increased efficiency; this would negate part of that.

To do it right would one remove the header, jack-studs, etc and put full-length studs in? Is any additional support necessary in the process of removing the header and adding new king-studs? Or just pop it out and hammer in new king-studs?

No matter which way you do it, you'll still have thermal bridging.
The only way to stop it is to build an exterior wall and an interior wall and stagger the studs so that they do not touch.
For example, if it's a 2x6 wall you would use 2x4s and stagger the studs on the exterior and interior.
That's a lot of work for just a door and window and I don't think you would see much of a return for such a small area.
I would close them up by adding the necessary studs, and insulate it well.

That's why we wrapped the house in foam sheets. The foam helps drastically reduce the thermal bridging. But doesn't answer your question

My house is wrapped in 6" thick rock... foam sheets are out of the question. I do have 2X6 walls with R-19 insulation though... I just figured if I have the wall torn open (all the drywall is already ripped off the inside and the rock is torn off the outside) I may as well do it right and get rid of what thermal bridging I can...

Is it off the wall thinking to rip the studs to 5", apply 1/2" of foam and then sheath?

It's definitely outside the box...

I'm not sure it's really practical though. Sounds like a LOT more work than just removing the headers/jack-studs and replacing. In the grand scheme of things, as pointed out earlier, it's not "that much" thermal bridging compared to all the windows and other headers etc in the house...