Allowing players to lock/unlock their own thread could prove useful. It would be less work for moderators, though minor, and therefore support.
No support. I don't think there is a way to implement this without letting you lock and move other people's threads. If they made it so you couldn't move it there would be a bunch of locked threads that need to be moved which would make more of a mess than if a moderator just locked and moved it. Also this could cause problems for when we lock threads for failure to adhere to forums rules and responsibilities people could just unlock and move them back. I never see this happening and I don't support, sorry.
Cyp could find a developer (possibly @Matt) that could create an XenForo plugin which applies a "player lock", which can be controlled by the thread's poster and the Staff Team. The plugin would also move the thread to the appropriate archive and moderator locks would not be able to be touched by players. If deciding to argue against this, I can see you making the point that this could be a hassle to code. I'm not that big of a developer but this does not seem like a hard project; regardless, I understand your words.