> But why doesn't player three joystick / controls register? Does JAMMA connector not
> support 3 players?
You got it in one. Just google JAMMA.
It supports 2 players. As yours was a 3 player cab, the relevant connections for the 3rd player controls is/was probably a seperate plug on to the PCB. Just trace the wiring from those controls and see where they go.
As to your other symptoms, I am wondering whether it is not a sticky switch causing it but rather a sloppy one resulting in switch-bounce and in certain games triggering a double-tap condition.
ie when you press the button, the switch, rather than changing from off to on, is actually going off-on-off-on (repeat off-on for number of switch bounces) and the game is then registering it as the button being pressed rapidly in quick succession and reacting accordingly.