The documentation for remote control focuses on running clients and servers on the same network. I see a few mentions of setting up port forwarding to make remote control work over the internet. Is there any official guidance on setting up remote servers to be used over the internet? How reliable is it in that setup?

There is technically no restriction but you need to set up your port forwarding to allow the connection over TCP for the port that you specify as the server. It should work fine as long as you don't go absolutely crazy with amount of data sending (like continuously clicking buttons as fast as you can etc), I'm sure you can break it if you try but under normal circumstances many of our clients are using it over the internet just fine.