SUCCESS!! 
(well, partially, anyway)
Issue #1 => SOLVED.I took the repair to the logical conclusion and replaced a bunch of ICs in the sound generation circuit. I decided to start with the readily available ICs first. If that didn't work, i would change out the RAMs, and then finally, change out the custom ROMs. I am happy to report that i didn't have to go that far! This is what i did:
1) Gave the mainboard PCB a thorough cleaning with alcohol. Lots of scrubbing (with a medium brush), and washed it down a few times
2) Removed the following ICs, installed sockets, and installed new ICs.
- 1L (LS237)
- 2M/N (LS174)
- 1M (LS283)
- 4K (LS157)
- 3M (LS158)
Like i mentioned before, since these are all commercially available, this was the easiest step to do first.
I'm happy to say that the sound issue is solved! The high-pitched version of the sound is completely gone, and has been returned back to the normal, pleasant Dig Dug sounds.
Now since i did them all in one fell swoop, so i don't know which one was the culprit. But since they are now socketed, if this issue ever reappears, it will be easy to replace one at a time, and i will definitely update this post with that information.
The humming (Issue #2) is still there, and honestly, its not really that big of a deal. But the above fix confirms that the problems are not related. I would guess that it's pickup from a noisy ground, or something like that. I'm willing to live with it.
The BAD NEWS? Dig Dug is STILL BLUE! (Issue #3)
So, if anyone has any suggestions and knows which specific ROM or RAM controls the colors on the individual sprites, i would be most grateful for that information!
Thanks.