First what I think is your easy fix. Pop bumper is probably a bad/missing capacitor. The cap helps eliminate switch "bounce" so if it is bad, the pop bumper can energize more than once. Could also just be a switch adjustment.
Now for the switch issues. Switch matrix issues are tricky. First, since you did some work on the start button, make sure that it's fish paper on the switch is properly preventing the switch from being grounded. If the fish paper on the credit button isn't there, you WILL have issues with the switches.
Next, go to the switch test in the diagnostics. From there you might find a pattern of all switches on a particular row or a particular column are giving you problems.
Next, visually check the diodes on the switches, including the pop bumper. A bad diode will cause matrix issues. Note that you can also test each diode with a multimeter, but to do it right, it takes a lot of time. You disconnect one end and then in diode setting test the the diode both directions. One direction you should get null reading, the other direction .4 - .6. So I wouldn't do the multimeter test to start with. We're trying simple stuff first, so just visually inspect the diodes on the switches for damage or being disconnected.
The difficulty is in determining whether your problem is in the wiring or on a component on the board. To determine that try these steps:
Make sure game is off and turn off all MPU DIP switches. Go to switch test. With this running, disconnect MPU connectors J2 and J3 (switch row and column signals). Connect an alligator clip to J2 Pin 1 (this is column 0 of the switch matrix). Then touch the other end to J2 Pin 8-15 (these are the rows) one pin at a time. Each time, a switch number should show in the displays. It should correspond to the matrix as diagrammed in the manual. Then touch the second end to pins J3 9-16.
Once you have done that, move the first clip end from J2 pin 1 to J2 pin 2 (column 1), and repeat everything above. Repeat this for J2 1-5 (all columns). This will work its way through all columns and rows.
From the above, if a row or column is bad, check the diodes on the switches for that row or column. Also, check the connector for that row. Again, you will need to compare this to the diagram in the manual to know which switches are on which row and column.
If you still can't find the problem, it may be PAI U10 (sends switch data). To test, you can swap U10 and U11 (hopefully they are socketed) and see if this clears the problem.