Shift Grind: 1-2 (Cold), 3-4 (Cold), 4-5 (Anytime)

I'll take a shot at this. Don't worry about reverse it's not synchronized (for reasons that should be obvious) and will have a tendency to grind, so before you shift into reverse stuff the shifter into a forward gear (synchronized) and this should halt most of the moving parts, then go to reverse.
However, it would seem that the clutch may not be adjusted properly and could be dragging, this would cause the input shaft to move, one other odd thing that I have seen is the pilot bushing/bearring go bad and also cause the input shaft to spin. If I'm not mistaken I think this trans has the "automatic style" friction material instead of the brass synchro rings, this could be wearing off (you didn't state mileage) This sounds like it has some posibilities since you stated that temperature seems to be an issue. You haven't changed tranny oil letely, and if you did, was it replaced withthe same type/weight etc,etc, and BTW the trans does have fluid in it..right?

Yeah, I wouldn't worry about the reverse too much, as it seems to be common. I usually just put the car in 3rd, then shift to reverse, and this makes it much easier. Also, I was reading my owners manual, and it explained something about a "reverse lockout". This is where it will not allow you to shift to reverse, you must clutch out, then clutch back in in order to engage reverse. Also, I (like most TR3650 owners) have the cold 1-2 "clunk/grind". This is an obvious problem with our cars. I did the tranny flush, and it seemed to help. When was the last time your tranny fluid was changed??