Same protocol at Cambridge (UK), at least in the rooms with two doors (outer door open: come in, outer door closed: do not disturb/I'm out). Lock picking wasn't really a thing, but one of the tutors did point out to us that the locks in a specific corridor could be opened by sliding a knife down the edge of the mechanism (it was "fixed" by screwing a steel plate to the _outside_ of the door, which felt like missing the point somewhat).
I'm possibly missing something obvious, but could you explain why you'd need to pick a quadrant? I'm not seeing why that is necessary.
If you don't care who else uses it while you're in there, dial "don't care". If a man dialed "women" and went in, he'd likely face some angry students. To my knowledge that never happened.