My knowledge of sound cards is fairly limited, so I couldn't really give you any information that you probably don't know already. As far as a 387 goes, I know that the 386 was the last generation to really use a genuine co-processor; later on, with the 486, it was just a 486DX instead of the onboard 486SX. I'm not sure exactly what a math co-pro is good for, but I remember seeing it used on Computer Chronicles at some point. I think they were using it to accelerate Auto-CAD software. It might have uses beyond things like this, but the impression I've been given of it is that it's more for business users who want to do things that require a heavier "3D" workload, or something along those lines. It might speed up general use a little bit, as I'd assume it takes some workload off of the CPU, but I'm not sure how much it would free up on the CPU load side of things.

I would take a look at that Dallas clock battery, though. They have become sort of infamous for failing, and being a general pain to replace/fix. If yours is soldered to the board, I feel for you. There's some people that have managed to install a hack that acts like a Dallas clock battery, but is in reality a CR-2032 battery with the recharge part of the Dallas unit disabled; I'm pretty sure that a mod like that only works with the socketed versions of those things.

If there are any expansion slots free, you might be able to get an external SCSI/early IDE controller that can handle larger HDDs. Getting a new HDD can be a tricky thing, as you don't know if it's going to fail after a week or a year, especially with how the early 90's ones tend to age. If you know that your current ~600MB one is good, I'd try and see if an external controller wouldn't be a better bet. And if your ~600MB HDD ever fails, you should have a higher ceiling to get a larger, and in effect, cheaper and more available replacement with the external controller installed. If it's a SCSI card that supports more than one device, or daisy-chaining, you might even be able to install the CD-ROM on the same controller as the HDD.

Just booted the system up with a MS-DOS boot disk. Running Speedsys as we speak.
Yes, the Dallas RTC appears to be dead. Entering the BIOS system setup menu shows every field with empty values. I've ordered a replacement. Fortunately, it appears as though it is socketed.
This system just seems so well designed. It has 2 MB RAM in Bank 0 soldered directly to the motherboard, and 6 30-pin SIMM slots each filled with 1 MB for a whopping total of 8 MB.
Speedsys shows my WD Caviar 2635 drive with the correct name, but I cannot imagine the BIOS actually being able to recognise the size. I'll have to wait for the new Dallas chip to see for sure though.
Then again, Speedsys also claims the CPU is a 386DX, rather than SX, so I wouldn't trust anything it says too much.

Today, I have finished my 386, at least until I make up my mind about needing a CD-ROM drive or not.
Specs at the moment:
386SX 20MHz + 80387SX
8MB RAM
1.6GB Quantum Fireball ST (using EZ-Drive for full size)
Guillemot Maxi Sound 16SE (ESS1868)

Should be plenty for playing older, speed sensitive MS-DOS games.
And I also have the nice and correct mechanical keyboard that came bundled with the system back in the days. Had to give it a thorough clean before I felt it was okay to handle though.
I am very happy with how it turned out.

Nice system! I have the Canon C-200 M53 which uses the exact same motherboard and similar case design. Currently trying to find MR. BIOS for it to see if it allows ROM shadowing. I use XT-IDE rather than EZ-BIOS, and that needs shadowing to perform well. What BIOS version does your board have? Mine is 1.2R3.