1) I use a false bottom in my sankey MT, and have always done so. It still has the occasional stuck sparge when using some oats or high-wheat grists. (I probably crush too finely, but that's beside the point) Stuck sparges in a sankey are a PITA, and you really need a good mash-paddle to get the grain back into suspension.
I would guess you'd have more stuck sparges with a screen, and that would be no fun. So, I'd suggest going with the false bottom. You'll probably be able to drain faster at any rate. Anything to speed up those 10 gallon brewdays is a plus.
2) Every false bottom for a sankey that I looked at went the full-width of the keg. They all folded in half to accommodate the cut opening in the keggle.
Mine goes under the valve opening, with a dip-tube that goes down through the false bottom. I can't see any other way to do it other than a bottom draining tun, which I have no experience with.
3) Thermometer. dunno. I check mine at dough-in by stirring wildly for 5 minutes, and swirling a thermo-pen around the surface. After that, the insulation keeps it at temp for the mash. I'll re-stir it afterward, and check the temp. (usually lose a degree or two,)
Good luck with the keggle-MT. It's a great way to mash. Make sure whatever insulation you put on is removable, wort getting down in there can turn nasty.