It's an AC motor, they can start in either direction. Wires are fine. Inside the gearbox is a lever that blocks gears in reverse, so it starts in forward. Grind rivet heads, open gearbox, clean out old grease and regrease. Reassemble w JB Weld.
Don_C

An AC motor does not care how the wires are hooked up. You can check the schematic to confirm they are AC wires (colors are right). You can also confirm if it is AC with your multimeter. When set it to DC, you will not get a stable reading, when you set it to AC, the reading should be stable.

I see the motor/ flywheel cover is missing in the photos, but it's not essential. Check the cabinet.

Yes, there is a risk to opening the gearbox. But what good is it to you now? There's no way to fix it without opening it. I have done this twice on gearmotors for pins, and several times on fans and a Miller Beer light. That being said, the patient may not survive. Other things may be worn out. I can't guarantee that I have diagnosed your problem correctly, but I am sharing my experience (from memory, I did this years ago).

Use a sharp drill bit or a Dremel type grinder to remove the ends of the rivets that were mashed to hold them in. Take the case halves apart carefully, then take some pictures. You can take out the gears and you can look for the little moveable lever that should make the gears turn only in one direction. Use some solvent and clean out all old grease. Make sure everything moves freely. Regrease lightly with some white lithium grease. Put all the gears back in and assemble the case halves and rivets. Using solvent, clean any areas that you will be gluing. Put a dab of JB weld or other quality epoxy on the ends of the rivets and the the case around the rivet holes. You can also put some epoxy on the seam of the case in a few places to help hold it all together. Let it all set up. Reassemble and try it out. I think mine still gave me some trouble to start but settled down after a little while. It's been working flawlessly for over ten years after the repair.

Good luck and let us know how it goes.

Don C.

Congrats on solving problem #1.
Coils: Be methodical. Check that they are free mechanically. With the game on and a game started, check (DC) voltage from either coil lug on one of the coils to ground (the braid). Should be about 40 volts. If no voltage, its connectors/wires. If you have voltage, then use your manual to find the driver transistor for one of the coils. Game on and started, briefly ground the tab of the transistor to ground. If the coil fires, the wiring to the coil is good. If not, next suspects are connectors, header pins on board. Then check the transistors themselves (search Pinside). If all OK, go up the chain to pre-driver transistor, then the buffer chip that controls the coils. Don't have a manual handy, so i don't know if they share a buffer chip.
GI: Cyclone GI circuits are overstressed. Take your meter on AC volts setting and see where you lose the voltage. Most likely you will find a fried connector/pins on the relay boards, driver board or even a wire-to-wire connector. Brown and crumbly connectors are usually the clear sign. If you find some, you MUST replace the connector and the pins together.
Don C.