Do devices have to be connected to Internet to log in and use locally

Plex is dependent on the internet.
Many client types can handle temporary internet outages fine, as long as you’re not attempting to switch user during an outage.
But for initial setup, internet connectivity is very much required.

And frequently moving the server machine into a different location is a world of hurt. Don’t do it, unless you create a whole LAN (with its own router) that you can take with you.

I think your use case is better served with an Android box, like the nVidia Shield and installing Kodi on it.