1. yes, the container pulls afaik the new update from the AUR automatically. You only need to update the container and minecraft will update itself
2. there are two ways to update.
First use the built-in function with a cronjob in the userScripts plugin
docker exec -u nobody -t minecraft /usr/bin/minecraftd backup
this would automatically backup your minecraft server, but u need to use the restore function to restore it because it changes some pathes.(Edit: after looking into the script it seems like it does not change any pathes so you could also copy it instead of using the restore function)
Second possbility is to stop the container and copy the whole minecraft directory and copy it back if needed