House Door: How do I take this apart without destroying it?

It's pretty much been covered here, but I would consider getting the glass out as opposed to just scraping the paint.  I mean... try one to be sure, but I would think that the edges where it disappears under the bead might just be a mess to get cleaned.  There are fewer things more frustrating than trying to remove old finish from millwork's nooks and crannies.

I would personally remove the bead on one side with no regard for it's health.  remove the glass, scrape it, coat it with a vinyl frosting, remove the pane stop (bead on the other side) strip the door, sand, and buy new bead to replace the stuff you took out.  You can likely find matching bead at the orange or blue stores.  If you can't find matching, I wouldn't give a hoot because you can't see both sides at the same time.

I also like to set glass with a tiny bead of putty caulk, glazing, or a tiny bead of silicone.  That way when you close it, turn up the TV, or someone walks by and shakes the floor, you don't have rattly glass in your door.  I have also done it (for theater) with a strip of 1/4" spike tape on the glass edges, but up close you can see it.

If you can swing it, get yourself an 18 gauge brad nailer for re-installing the beads.  swinging hammers near glass is a recipe for disappointment

It turns out the dumb  thing does not actually have glass in it.  Those panes are wood.  I'm going to strip this door down and refinish it and if anyone wants to come over and help they're more than welcome to.

I believe that any delicate work should be successfully completed when a person isn't in a hurry. If you do everything slowly, you'll be able to remove all the slats from the door and insert glass there. I have oak internal doors with four panels installed at home. Of course, I've never tried to change these panels since I'm satisfied with the pale blue frosted glass. But after reading your post, I decided to take a closer look at what exactly binds the panels of my interior doors with glass. There are very small screws that aren't visible to the eye and almost merge with the dark color of the wood. If I were you, I'd pick up suitable tools, screwdrivers, small scaler (dental instrument) that allow you to make small movements even for jewelry work.