I don't think I would cut a hole in the door, rather go to the space just to the right of the door, which looks like about 6" or so, and run your ducting out there. You could probably cut a hole for you intake air in that same area. If your trying to go stealth, you'll probably have to keep the intake and exaust down low and just cover up your duct hose. If not you can run the intake down low for the cooler air, and exaust up high to create some air movement in the room.
I'm throwing these pictures in just in case they might be helpfull for you. I'm working out of a closet also, which I'v turned into two seperate rooms.