You call out of libgdx into custom java methods. It would take some work but i'm sure it's possible. Take a look in the Android SDK in the demos folder for demo-realtime and hook up a way to call out of libgdx into the hosting android Activity. I posted some code here that you can use or any way you prefer. It looks like the Intents they are launching are just to show other views (Help, Options, etc) that you can definitely do if you want to jump to different Activities in your game.
If you need to call back into libgdx, use the Application.postRunnable method.
Yep, that's what I would do because every time you leave the gdx hosted Activity and return, it will cause the gl context to spin back up which may lead to a poor user experience. If you want to have different views/menus hosted in your Android app, use something that doesn't leave the current Activity like this:
This window can be launched from a method in your Activity class which you could call from libgdx and you won't lose the gl context in libgdx.