How does the laser printer operate?
The paper is generally picked up by a rubber roller(s). Try cleaning them with a damp cloth. Make sure there are no small pieces of paper in the paper path.
Sounds like something in the lock mechanism or linkage is either worn out or sticking. When the b/f puts WD-40 on it, does that fix the problem for a while? If so, it is probably sticking maybe try using something thicker than WD-40, which really just drips off or evaporates after a short while. White lithium grease can be bought in a spray can in most auto parts stores and is much thicker than WD-40 so it doesn't drip off. If that doesn't work, you may need to get a new door from a junk yard.
Absolutely. I have worked in several productions in both church buildings and theaters, and the fire alarms are always an issue. They can't tell the difference between manufactured fog and smoke.