The standby server regularly performs restorepoints, which are checkpoints in standby mode. On startup, it checks its control file for the lateat restorepoint and recovers from there. After it has completed replaying local WAL, it connects to the primary and asks for WAL from that point on.
Shut down both servers in any order you like. If you ...
Mirroring has been deprecated for a while - I wouldn't build out things with it at this point.
I think you may be better off spending time preventing issues that cause you to have to re-do your replication, but aside from that note I'd recommend looking at using Availability Groups instead of, or with, transactional replication.
For using with transactional ...