Prop and coupler both need to be fixed so that neither can move without you using a spanner on it – locknuts AND thread locking compound are both good. A prop unwinding itself is deeply annoying. So is a prop that manages to tighten itself up, because then it acts like a very effective brake. The usual follow-up question is along the lines of "Why has my motor/esc/wiring let the magic smoke out?"

Yes you need thrust washers to stop either end wearing the prop shaft bush faces or the prop or coupler face. When going forward, the prop is pushing the outer bearing forward, n reverse, it tries to pull the shaft out of the boat.

You don't want threaded bits of shaft inside the bearings.

No end load should be transmitted to the motor. The motor bearings are not usually designed to cope with that, they just stop the inside bits from wobbling about and allow the motor to turn.