That's a lot of information. Some of those things may be problems and some may be related, but trying to untangle them all remotely makes it more complicated than it needs to be. I would recommend trying to isolate the individual problems so you can easily reproduce them. Isolate, simplify, and minimize the number of variables.
Also as a general rule, before you adjust switches to get them to operate properly, check that the screws holding the switch in place are tight. Otherwise the switch can wiggle into a different position and out of adjustment again. For extra credit you could go through all of the relays and steppers in the game and operate them all manually with the power off and visually verify that they all open and close as they should, the switch stacks are tight and the lugs aren't shorting. This often finds a problem or two before you even know it's there.
So you mentioned that a Hold relay switch was out of adjustment, and later mentioned that the Hold relay isn't firing. That's one problem to isolate and diagnose. The Hold relay may be fired by the Start relay. Until you get the schematic you could try to trace the wire from the Hold relay solenoid to the next switch to see if it is operating correctly. If you can't follow the wire color, you could use a meter with one probe clipped to the Hold relay solenoid lug. With the other probe touch all of the other relay switches in the bank looking for a switch with a resistance of less than an ohm or two.
As for the pop bumpers, experiment a little to see if they always stop scoring at the same time. Maybe start with just one pop bumper and understand how it behaves. What changes from the time that it does work to the time that it stops working? Are the pop bumper switches clean and operating properly?
I don't know how the game goes into game over mode, but I suspect that both of the ball trough switches need to be closed indicating that the 4th and 5th ball have drained. Is there a game over light in the backbox? If it's not working, fixing that first might help solve other problems later.
And if the game isn't starting be sure that there are credits on the game, and that the switches on the credit unit are operating correctly. There are one or two switches that close whenever there are any credits on the game. They open when the credit unit shows 0. Make sure those are working correctly.