Sounds like you have a few problems. Some of the easier ones (that will help narrow down behavior):

1) The lamps for each set of three cards are in series. In other words, if you hit the number 16, it will light the number 16 on cards 1-3 on one circuit and 4-6 on another. You have a bulb that is either burned out or not making good contact on one of the cards 4-6 for whichever numbers are giving you trouble. Note that on Nashville, the card lamps should be 47. This may seem like a trivial thing, but the circuit is designed for that resistance. Other lamp values will cause weird problems.

2) Slip ring wipers - yes, they should alternate. If there are five on one side and four on the other, you would start on the edges with the 5 side and alternate. The manual, as you can see lists the wire colors as well as the wire color code as shown on the schematic. If ever in question, you can trace that specific wire on the schematic to the next component and use a meter to test continuity. Generally speaking, placing two wipers on a single slot will cause fuses to blow.

3) Yes, on most six cards (including Nashville) all search discs should start at 0. If they are off of zero, you'll get a wild 'christmas tree' lighting effect, and phantom hits. The manual shows the zero position with a number of connections (the little arrows pointing to the rivets - these correspond to one of the wiper fingers). You can use the other little arrows to verify that the wiper arms are in the correct orientation. Ensure that that switch stack on the edge is lifted only when the wiper arms are at zero/right before zero. There's a small amount of play in the switch stack position via slots in the bracket.

4) Replay register rolling back credits when won automatically means you have a 50V sag. 50V runs (almost) the entire game in a Bally bingo. If you have a temporary drain on that line, the game will 'reset', which would cause the replay reset relay to trigger. It's also possible that that relay, the replay register switches, or a start relay switch are out of adjustment.

Just a note about normal operation - when searching, the 1-2 search disc will roll to a stop at zero position, which then triggers the coil to release 3-4, and when that rolls to a stop it'll release 5-6. If that sequential behavior is not happening, something is out of line. Either the limit switches triggered by the pin, the switches under the C button, or the discs themselves. Nashville uses the plastic couplers on the search discs. If they were assembled incorrectly, there would be a lot of play. Pressing the latch relay plate down and spinning in the correct direction, you can feel the difference between 1-2 and the others if you suspect this might be the case. To fix, if that was the issue, you'd have to disassemble the search disc 3-6 unit and place the wiper arm 'collar' back in the bushing. You can place it in upside-down, and outside of the intended holder. Verify position with your manual and you'll be fine. That unit is a little tough to disassemble on a Nashville, but patience will get it done.

Yeah, the instructions I posted about the plastic couplers are apt in this case then. Alternatively, the clutches have dried or been lubed with something that turned into glue and the shaft can no longer turn - OR - the motor has burned out because of sticky clutch washers. Definitely, with power off, try spinning the fan on the motor and see if the shaft spins as you turn. If there's a huge amount of resistance, you've found your problem. You'll have to disassemble the search discs anyway to change the washers, but this is a once-in-many-years type maintenance in a home environment.

OK, yeah, you surmise correctly. Alternatively, Nashville has diodes on the win/lose relays on the trip bank on the back door. On Dixieland (the next game), Bally undersized those diodes for the application. They're fine until they're not. I worked on a Dixieland once where every diode had to be replaced. Anyway, the symptom of this would be automatic double-or-nothing play and always receiving nothing without any user input after pressing 'C'. Alternatively (because there are always multiple possible solutions) it could be a misadjusted switch on a particular card's trip relay(s) or a fault in the search index for the search disc in question (which stops the search disc and has to activate on a hair trigger).

Regardless, aligning your search disc is definitely job #1.

You'll have to either fabricate something (may be a 3d printed part in a rigid plastic?) Or pull from a broken game. The search wiper configuration varied from game to game, so worst case you'd have to rivet a finger into an alternate position and/or do some soldering. The search disc wiper arms from any of the 70s six cards -should- work, theoretically. At least for basic wins.