What type of board is it?
If it is a Core business board, then issues will disappear 14 days after they have become "done". If you are in a Scrum board, done items will go away at the end of the sprint, and on a Kanban, when you do a release (although if you're using Kanban and don't do releases, you might want to just change the board filter to add "and not (status = done and updated < -7d)")
Ouch. Moving issues between projects is a really bad thing to do as part of a day-to-day process. It it good for re-organisations and housekeeping, but it should never be part of a working process.
May I ask what is wrong with "seeing 'done' for 14 days before the issues automatically drop off the board" for you?
No, your definition of "archiving" feels a bit wrong to me. I archive things when I have confidence that they don't add a lot to immediate reporting, but might need to be kept for statistical reporting or audits. The shortest time I've seen for an archive is "three months of inactivity"
Dumping "done" stuff into an archive just because you have done it this week sounds very wrong. What are you telling your stakeholders here? That you have binned stuff just because you think you might have done it?
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