The working assumption of most scholars is that resignation or the end of a term stops the impeachment proceeding. Otherwise, with the holding of office no longer at stake, a ban from future office or the forfeiture of pension become the only punishments available. A trial before a legislative body for such a purpose would likely be an unconstitutional bill of attainder.
Exactly. The core purpose of a senate trial to remove a functionary from office. All that other stuff presupposes the senate had jurisdiction over such a case to begin with, which it wouldn’t.