Let's "set the stage". You have a toggle switch and a LED, attached to an Arduino. Also an output signal going to the game interface. You just need to implement the suggestion in reply #5. Do you understand it? Do you have some specific problem implementing it? Here is some pseudocode:
loop
{
read switch
if current switch value is not the same as the saved switch value
{
toggle LED
send a pulse to the game
set saved switch value to current switch value
}
}
Considering the sync problem, there is nothing to "get your head around". You can't fix that without additional communication - either the game has to have access to the true switch position, or the switch has to have access to the game position. Also, a physical toggle can't be modified by software, it's position is determined by physics. So make sure that your pulses work well.