He will get up and close it himself actually. I open it when he's asleep though (there is a gate in front of his door) because our house is new and the dampers need adjusting and his room becomes a sauna.
I used to leave it open but the dog thinks that the whole house should wake up @ 6am. She also figured out that even though I ignore her I won't ignore A so she'll wake him instead.
Closed for naps and closed until we go to bed. We open it a few inches then, but mostly because its so cold in their room so I like the heat to even out.
This for us too. Otherwise the cat would scratch at the door and want in. If the door is closed but with a teeny tiny crack, the cat doesn't bother with it.