Alrighty.. I had a quick look at the supply board schem... Here's some notes that pop into mind:
When you look at the PS schem, break it down into three seperate parts.. If you draw a line across the page at about the middle, everything from this imaginary line and up is the 5 volts regulated circuit. You can pretty much see this circuit kinda stands alone from the rest of the diagram.
Let's look at the 5 volt first, as it's the most crucial to board operation. Let's measure at the PS board itself so it's easier to follow the schem. With your meter in DC Volts mode, place your BLACK meter lead on pins 4, 5 or 6 at the PS board (they are all the same, ground circuit). Place your RED meter lead on pin 8, 9 or 10. You should be reading 5.0v DC or so (sounds like you are from your post)
Now, leaving the black meter lead where it was, probe pin 7 with the red lead, your meter should read -5.0 v (give or take a little here is ok) Note the MINUS before the reading, this is the negative 5v circuit and should read as -5.0 on your meter.
Ok, moving along, let's look at the lower part of the schem. This lower part we can break down into two smaller parts, left and right. The left part is the 12 volt DC regulator circuit and the right part is the audio amp.
Starting at the left part, we'll check the 12v reg circuit. Again, with the red lead of your meter (leaving the black where it was from above) check pin 2, it should be 12v DC or so. This could be one or two volts higher or lower, which would probably be normal. Now check pin 1, it should read -12v DC (again, note the minus).
The rest of the circuit, on the RH side of the schem, is the audio amp. If there is a problem with the amp, it shouldn't really cause a problem with gameplay, just sound issues. Still, you should be getting close to 0v on the speaker wires with no sound occuring, so this does lead me to suspect your supply/amp board is damaged. Sounds like one of the amp transistors is shorted (the TIP120 would be the likely culprit). But again, this shouldn't be bringing the whole game down.
For now, let's focus on getting the board up and running and worry about the sound a little later on, I'd disconnect the speaker and also see the connection labeled "Audio In" on the PS/Amp board? Disconnect that too and focus on the largest problem for right now.
Post back with your voltage measurements as I described above and we will see where you need to go next.
As for your ROM question. The only way to test them (besides in a working board) is to read them with burner and compare checksums to good images. A burner is not something I'd recomend you get quite yet, it's not something you would use hardly at all (if more than once). But, I don't think your root issue is the ROMs. But at this point I am concerned about logic board damage from incorrect voltages. That's why I am curious what voltages you are getting on what pins of the PS board.
Regards,
D