As I have already said if the tank is full there is more than enough air to raise the car to standard height. No air is going in the tank.If the leak was in the tank side the compressor would run until it overheated which would be longer than 7 or 8 minutes.
From the OP's description the tank fills, the compressor stops until the door is shut when it restarts as pressure is lost when it tries to fill the corners.