Every modded 2836 I've seen come out of the factory has had the jumping GMT hand for at least the last 2-3 years. I doubt yours is any different. A 2893 will be functionally the same.
Probably not. If you uncase the movement, and remove the stem like normal using a pair of tweezers or pliers, it should just come out. No reason to think the keyless is messed up just because the stem broke.
Better question is why the stem broke--the usual reason is sideways pressure on the stem due to incorrect stem height for the case. If this is true, the next one's just going to break too.
It looks like just the stem broke. A new stem is ~$3.
Not telling you not to buy a gen crown if you want one, but it doesn't look like there's anything at all wrong with your crown and this is a repair that can be done in under 10 minutes with under $5 worth of parts.
Why would you use a super-sea-dweller case? It's entirely wrong for the project. Most people who build tudor snowflakes use the 1680 or 5513 cases as a starting point.
I have had some success fixing this by putting several drops of superglue on the inside of the bezel, letting it dry, then snapping it on the retaining ring and playing with it until it looks right. The superglue fills up the space and keeps everything tight. It's a crappy, ugly hack, but at least your bezel stays on.
The only "real" fix is to source another retaining ring or bezel ring that is the right diameter
(Note I am not saying just glue the two pieces together)
Yep. And in some ways, it's done better--the GMT wheel sits under a retaining plate, for example, so there's no issues with it lifting slightly and losing contact with its driving gear. However, in terms of concept, it's exactly the same--it's an additional wheel that sits on top of a movement which was not originally designed to have it, driven by the wheel that drives the calendar mechanism.
No, it uses the calendar drive wheel, which is present on both the 2824 and 2836. It's a slightly different part (on the 2824 it's integrated into the calendar plate cover thing while on the 2836 it fits on the mainplate itself) but there's no conceptual reason why you couldn't easily build a 2824 with GMT function in exactly the same way that the 2836 GMT works.
The day changing function is needed to adjust the hour hand though, so it would be non-adjustable if you did this on a 2824 unless you install the 2836 style double corrector.
I'm not sure why you're so hung up on this, but yes, a gen should click 120 times. Many reps do as well. Some older gens will not, quite, due to wear or damage to the bezel teeth and anything model 1680 or older doesn't click at all.
You'd have to manufacture/fabricate/source all of the parts that make the 3185/3186 different from the 3135. That's much of the dial side of the movement. Unless they decide to clone them from the factory, I can't imagine being able to do it for less than just buying a 3185 would cost.
So you're thinking of using an ETA dial on a Rolex movement?
Mind if I ask why? Usually people go the other way around, since the dial is the part you can see...