1) "Please keep this door closed."
The second, however, seems more difficult:
2) "Please keep this door closed-ish"
The door does not close completely, and it is designed that way so as to allow a cat access into the room at all times, but the door needs to not be swung wide open due to heating/ventilation problems, so any equivalent of "Please keep this door closed-ish /as closed as it will go/etc" will work.
Peace and thanks!!