Midway liked to use 6.3V lamps in the coin doors from that era. Should be the yellow and yellow/white stripe wires on that connector. They actually had a dedicated winding on the "power pack" main transformer (that also served as the monitor iso and handled step up/down if needed) on some models, such as MK2.
You can run it off the 5V from your PC, but the lamps will be a bit dim (probably fine). I replaced them with 14V lamps and run them off the 12V switching supply I have in mine (which is just an old ATX supply) with similar effect.
As for power, I just have a standard power strip bolted to the bottom. Not my ideal solution, but it works well in this case. I wired the existing top switch inline to easily turn the cabinet on/off. I don't generally have a PC in it, and when I do, I have the PC set up to be fine with being hard-powered, so that works well.