OK, let me paraphrase your situation. You have 4 combinations and have only reported on two.
You send a signal from pin11 to pin12, and use pin13/LED to verify it comes through. This works if the connection is very close (Why did you use a diode? What do you think happens if you reverse the diode? I think it will "fail"). You do not say if things work as expected if you do not connect the two.
This fails if you have long wires which go "somewhere", claiming there is a connection although your Ohm meter says they have no connection. (Somewhere is a switch matrix, or circuit or ?) Does it work as expected if the switchmatrix-circuit-whatever is closed?
The observation that if you wait "delay(1)" between setting pin 11 and sampling it on pin 12, it yields correct result (you only say on the open, not the closed state). Have you tried delayMicroseconds(100), (50), .. (four is the minimum)? You code without the delay will turn the pin on and sample it on/off the other approximatly 300-500 nanoseconds later (I have not worked out the precise timing) or 15 meters at lightspeed. Any capacitance/inductance will drastically reduce the speed. The pinball components work in deci or centiseconds for most stuff. Your Arduino is working in sub-microseconds.
Lastly what makes it "scan one set of switches on LOW and another on HIGH"? A circuit? Diode? Relay?
Good luck on the continuing bug-hunt 