If you're not getting any connection to the backbox, I don't it's worth looking at the reset issues yet. Start by trying to figure out what's happening there.

Things to check:
1. Check Jones plugs in the backbox and on the mech panel under the playfield (the bottom of the cabinet). See if any are disconnected or damaged. (If the head was removed for transport, then these might not have been reconnected properly.)
2. Check the bundles of wires between the backbox and the mech panel. Do any of them look damaged?
3. Are there any fuses blown or not seated well?

If you don't have a printed copy of the manual and the schematic, I'd recommend ordering one from http://www.pbresource.com/. For short-term reference while you're waiting for that, there are PDFs of those here: https://www.ipdb.org/machine.cgi?id=1733

The far right of the schematic (top if you're looking at the PDF and haven't rotated it) shows that you have 5 DC coils (3 pop bumpers and 2 kickers / slingshots). The majority of the coils and all of the lights are running on AC. This seems to be typical of mid-1970's Williams. I have a 1975 Triple Strike with the same arrangement (except it only has one pop bumper).

Even if the colors are faded, you might be able to figure out what is what with the help of the schematic and using process of elimination.

WARNING: Don't use any kind of contact cleaner. They are highly flammable! (You didn't say how you would clean, so hopefully that's not what you meant.) Switch contacts should be cleaned with flexstone or a file (depending on switch type). As recommended by a previous post, pinrepair.com has some good info about how to do this.

WARNING 2: It sounds like you're talking about doing a "shotgun approach" to the problem. Trying to clean / adjust *everything* risks creating more new problems than you're solving. This is particularly likely if you're new to EM pinball maintenance. I'd recommend focusing only on things that are potentially related to the problem you're trying to fix.

If it was me, I'd focus on problems in this order:
1. Getting power / connectivity to the backbox working completely and reliably
2. Getting startup / reset issues resolved
3. Issues that affect a smaller number of things but prevent full testing of feature and / or gameplay (e.g. a switch is stuck so none of the 10 pt scores work)
4. Issues that are isolated and don't affect other things (e.g. a single switch that doesn't score points)

I think they're talking about the tic-tac-toe squares. There's a path from "1-O" to "2-O" to "3-O" relays that will complete when all three relays are energized that will trigger an "O wins" condition in the "1-2-3 combination". (I don't know OXO well enough to know what a win means, but it's tic-tac-toe so I know it's trying to complete a line of 3 "O" or a line of 3 "X" in any direction.

That loose wire looks like it may be the same color as one of the wires soldered to the A-22 550 coil nearby. Maybe it was soldered on the same lug and broke off?

The closeup shows two black wires on the far lug and what may be black with orange on the middle lug. The near side has no lug and no coil wire running to it.

If you have a multimeter, it wouldn't hurt to test resistance across the coils. This chart should tell you what values to expect: https://www.flippers.com/coil-resistance.html

Don't worry if you don't get the precise values, but if it's really low, the coil is shorted out. If it's really high, there is a break in the coil wire somewhere.

From what I see in the schematic, I think ball index relay will hold until the outhole relay is triggered (end of ball), but I don't have an OXO to check the behavior. I would check the resistance on that coil and see if it registers normal or if the coil is starting to fail. You can also try pushing on the relay arm (preferably with a non-conductive item (wood dowel works well) and see if it buzzes less when you're holding it closed.