Well I did it.... got all three ticking time bombs out! It’s important that first order of business with classic Macs is to remove the PRAM battery! Leaving them in and or replacing them and forgetting them can destroy your classic Mac!!
I swapped an LCIII board into a LCII case several years ago and I don't recall there being any problems with that. FWIW I want to say there's enough spare room that the board's shorter clearance to the fan isn't a physical issue (the fan is held in by clips to the left and right relative to the above pictures). Presumable the other direction (LCII board -> LCIII case) is also amenable, given that the boards are essentially the same in dimensions and components as well as the side notches.
you can see the extra space near the fan here:
My guess would be that you could probably migrate a Macintosh LC board forwards with some caveats relative to hooking up various components and a LCII might fit into the LC chassis. It does look like from the below page that the LC had a larger fan+surround design that might not permit fitting the LCIII board, though and the total lack of the edge connectors might be a problem.
When my SE battery exploded, many of the traces that were sound-related around the speaker terminal were damaged. I'd check them with the multimeter and make sure there was no damage.
As for the battery, after looking at a few boards, I realized that they put in the two different spaced battery terminal through holes for different sized components. you can use any combination of two to make a connection (as you did).
Lastly, that orange crud did come off my board with some real work.It was in exactly the same place. It was hiding some rotten sound traces that run directly under the battery holder. I would focus my efforts here to find your sound issue. You can see in my photos:
Those MiniScribe 8425 drives are usually pretty reliable, though they're slow. They tend to make a lot of drama out of the smallest disk access, which is amusing, and they occasionally run a thermal recal routine which results in some clunks and peeps from the drive even when it's idle.
There's one problem with them, though (common to most stepper motor drives): they don't auto-park the heads, so when the disk spins down, the heads land on whatever spot they're currently over. Not good for your data, especially if the drive receives shock in this state. It's best to use a driver utility such as Silverlining and enable the head parking feature, which automatically moves the heads to the innermost track before a restart or shutdown. This causes this particular model of drive to make an unsettling BRRRRRT noise at the next spin-up when the stepper motor hits its innermost limit during the startup routine, but that doesn't seem to cause any problems and it occasionally happens on its own anyway if the heads land on some of the inner tracks between power cycles.
Occasionally the drives will suffer from corrupted EPROMs which will render them useless even though the media is still ok. They can sometimes be resurrected if you find a copy of the same version ROM and re-burn the chip.
That's a MiniScribe stepper motor hard drive - they tend to get a bad rap but make a set of sounds like none other.
If you can't get HD SC Setup to recognize it, try Lido instead - I had a couple of those drives that I'd all but given up on. Lido worked when everything else failed.