Yes just rsync should work if you are stopping everything on the first system. The second system should be completely switched off. If you can just mount the drive, that’s the way to go.
No, don't think so -- I would be going from one digital ocean VM to another. So the target machine will be in operation. I honestly might just leave it and think of some other use for the server to justify keeping it running in current state.
Thanks for explaining this though - really appreciate it.