So many options. As others have mentioned, rsync, borg, restic, etc. You might want to look into filesystem snapshots. If you use something like BTRFS you can create instant snapshots and send them to a second BTRFS formatted disk or even a remote system with a BTRFS filesystem.
ZFS would also work here.
I use btrbk for automatic BTRFS snapshots and backing them up to remote systems.
If you want built-in encryption you can use Borg or Restic, which also has the advantage of deduplicating within a single backup set. Restic can also backup to an s3 bucket, in case you want to use a cloud service.
Yes of course. On the btrbk homepage they even describe how to set it up so that a backup gets triggered automatically when you plug in a designated backup drive.
My setup is to create local snapshots and keep X amount of local snapshots. Copy snapshots to a remote server and keep a different amount of snapshots there. Finally I also have a backup drive and btrbk is setup to copy all my local snapshots to that backup drive when it's plugged in.