A883 2nd & 3rd question

Yes it could. Could be other stuff too. Like; shifter/linkage alignment/loose levers/worn combs/bad back bearing/driveline vibration/engine shake/tranny mount.
But the most likely thing for a no load pop-out is insufficient engagement, due to a very worn fork. The steel forks can easily be repaired(hard brass brazing). The cover can be removed from inside the tunnel.

The real problem comes from jumping out of gear repeatedly, as it ruins the interlock teeth on the gear and slider both.
That one is going to take some hard parts, as well as possibly some of the things you mentioned above.