OK, so there doesn't seem to be a special feature run by the electrics (although we haven't seen the front glass as yet to be sure). You say the machine runs and pays out. Are you saying you can put a coin in and the machine will play, or do you have to cycle the mech by hand outside the case?
As it seems from what we can see this is what happens: when you put a coin in it trips a switch which turns on the solenoid low down on the handle side. This allows you to play the machine; the solenoid stays on until the circuit is broken during the cycle which trips another switch (most likely on the other side of the machine as I can see some wiring there).
The hum you can hear is either the solenoid or the choke for the lights and both would be usual (I wont say normal, but usual after years of use).
The first thing to check if the solenoid is not firing is that power is getting to the coin mech. If so, move on to the switch the coin trips. Operate it by hand (using a screwdriver rather than your hand.
) See if the solenoid fires. Also check the switch that switches the solenoid off. This could be permanently in the off position, so the circuit never completes. You can check both these by bypassing them. It could be the solenoid itself. This can easily be checked with a multimeter to see if there is a break in the coil. Remember, electricity bites - even if there is a transformer in the machine you can't be sure the voltage is low in every area.
I think you said the lights are not working. Unless there is a break in the power line going to the choke, this would have to be the wiring, tubes, starters or choke of course. I always replace the starters first, then the tubes. These tubes are available on Ebay very cheap, often in packs of 10.