The capacitor is for powering the Alternating relay. I think in this application it's serving to 'debounce' the signal, so the relay can stay energized while the pair of switches settle their state, and perhaps also rectifying the AC signal to DC (although I've not researched that, but that's what rectifiers generally do).

The "Drop Target Sw." is the make/break you fabbed/replaced. It changes state when all four targets are down (C37 has two of these, as there's another 'drop target sequence completed' switch on the schematic, which is the one in the center of the target bank).

The diodes are there to only allow current to flow in on direction (towards the Alternating relay). Since you say that the Drop Target switch was getting hot, I'd guess that a diode was reversed, and the current from the cap was flowing back through the switch. The wiring in your picture looks a bit funky for what's left of the diodes, especially with the twisted wires, so it's at least not how it left the factory.

Replacing the diodes is fairly straightforward, but pay particular attention to the banded side of the diode for correct orientation.

I think so.

I'd suspect this is an 'improved' version of the other capacitor-based buffer they used on other games. Those have the cap mounted on the back of a Jones plug.