You are right about the coin switch; you should just wire it to your encoder like any other switch.
For the lights, my coin door lamps are 14V, so I just wired them to a regular pc power supply molex connector. The 4 pin molex connector on PC power supplies have two black wires (grounds) one red wire (5V) and one yellow wire (12V). Hook one wire from one lead on the lamp to the yellow wire on your power supply connector, and pick either of the black wires on the power supply connector to attach to the other lead on the lamp.
The big thing at the bottom of the cabinet is most likely an isolation transformer to reduce noise between the original power supply and the original monitor.