Does anyone know what features the LANC port supports, and what hardware can be used with it?

At the very least it will control rec/pause. I'll leave it to others
with experience using LANC on the C300 to elaborate from
here. Just about any LANC controller should work.

I've tested a Manfrotto Pro Lanc control with the C300 and it controls the 3 functions.

1. Focus - It pulls focus very smoothly and avoids the possibility of moving the camera as you change focus with your hand. the only downside ( similar to wifi/ iPad remote ) is that if you are close enough you can hear the focus clicking through.

2. It starts and stops recording

3. It also turns the camera on and off.

It might be that more functions are available with a dedicated Canon LANC remote.