I had that problem, also on a rear left door, and on a '96 318i - what a coincidence
It may be a door lock actuator being stuck.
As for the alarm, can you hear the honks when you click Lock or Unlock? If not even red light on the remote comes up - must be the battery. If the battery is okay - maybe, the alarm unit lost the remote's code - and the remote needs to be programmed again.