Yes, that switch should have a diode on it.

Doesn't sound like it, the leads go onto the tabs above the switch instead of the screws.
The diode you'll need is a 1N4004, they're very cheap and available everywhere.
Make sure the band of the diode is installed in the right direction, compare to other switches to confirm.

It's hard to say if the missing diode is the only problem, might be a failed switch or a broken wire somewhere else.
Test the switch on all three tabs while opening and closing it, compare results to a working switch with a diode.
For wire tracing, take a look at the switch matrix in the game manual, then test the wires from switch to switch.
If your not getting continuity where you should, that would be an issue too