Before
The same problem has hunted this car as discribed below. I ended up doing a clutch replacement and while I was at it, I did the starter (new) / transmission (good/strong condition) swap.
After
Starter
When I go to start the car in the FREEZING COLD (-22) it turns over the engine 1-2 revolutions, it then spins freely until let go of the key and ends with a dead crunch that sounds like the starter gear slamming into the flywheel.
Transmission
It didn't do this before but the reverse gear jumps out when there is a load on it and 2nd gear did feel all there but has slowly getting smaller on the shifter.
Car 1997 Volkswagen Jetta 1.9L TD
did you replace the starter bushing in the trans you swapped in?
is the starter in good shape?
The starter prob sounds like a weak battery/poor connections/bad starter/bad starter bushing. Reverse popping out/second gear feeling "shorter" in the shifter may be shifter adjustment. May also be the trans going bad, that's usually what popping outta gear means. Bearings get loose, gears cock at an angle and slide across each other and out of mesh.
You said you changed the clutch a while ago? ALL this could be loose bolts. Check all bolts, wire connections and oil levels.
The starter kicking out problem is a bad starter drive. While you are at it check the bushing. It is unlikely that the bushing would be the cause unless it is totally missing but it just might make things worse. Check the shifter adjustment as noted above. You might get lucky.