I think the problem is the switches.
If you can get a switch on each door which makes a *momentary* contact as the door is opened, but is otherwise open, you can wire them in parallel. They used to be sold for shop door-bells. In effect what you want is the reverse of a a SPDT switch with centre off, ie with a centre-on action midway between door open and door closed.
Could you do something with two reed switches per door, the magnets arranged so with operating distances so that when both switches are made (door closed) or both switches are open (door open) the circuit is open-circuit, but there's a gap in the middle when one switch is made and the other isn't that gives you a closed-circuit and a trigger pulse.
Owain