There are no capacitors on solenoids. There are diodes on solenoids, and ss bally's have small capacitors on the corresponding switches as well as diodes. If the capacitors are bad it might trigger a switch-closure, but that is only for that switch. So there will be no other solenoids firing because of it.

Regarding that U1 rom: "do not touch it". If it is working now, leave it as it is.
Without that rom, or when it has bad connections the game won't start at all.
If you want to replace it you will get another type of rom, and you will have to modify the pcb as well (different jumpers).

The diodes of the switches are a possible cause of your problem. Or it could be a short in the wiring of the switchmatrix. The slings, bumpers and droptarget all share the same column in the switchmatrix.

However that does not explain why some coils won't fire. It can also be that the MPU is giving the wrong commands to the solenoid driver.

I suggest you put the game in the testmode and go from there.( the red button in the coindoor)

- First the switch test. With all droptargets up, and no captive ball laying on the target, no ball in the outhole the switchtest should indicate nothing.
Then close each switch by hand and see its corresponding number.

- Then the solenoid test. Does every solenoid work from the testmenu?

Doing these tests are easy for checking if it is a switchmatrix-problem, or a solenoid problem.
If all solenoids work fine in the test, that means the MPU is controlling them in a correct way.

Very nice. That is a large amount of wiring/etc working perfectly.

That means the MPU sends the wrong signals to the 1-16 decoder chip.
Can be anything from U11 (on the MPU), connector J4 at the MPU, connector J4 at the solenoid driver, to the LS154.

Chances are that the connectors J4 (both boards) needs solder reflowing.

Could also be corrosion or a dirty U11 on the MPU. Not very likely, because the game is booting properly, but theoretically it is possible that one leg is not making good contact. In that case reseating the IC solves the problem.

I'd bet on the connectors, most probably the one on the solenoid driver.

In that case I would suspect the LS154 on the solenoid driver giving problems.

It's the 1-16 decoder (larger chip on the solenoid driver). What it does is translate an output based on a binary coded input. So "0110" activates output 6.
If you know how it works you can test it with a logic probe (all of the inputs should pulse depending on the output selected by the MPU).

But you can also replace it.

Not really. If only all acid damages looked like this....

If something sometimes works and sometimes not it's my experience that the solution can be found in bad connections / solder cracks.

And since you have an MPU with acid damage: I'd check for the traces on the MPU from U11 (pins 10-13, count from the lower/left side) to the connector J4.