Hello and welcome....

First for the soundcard, try another ISA slot. Does the Vibra 16c does have any jumpers? I'm not familiar whit that card. I use a Soundblaster 16 and I did not used the Creative tools or setup disks, Just plugged the card in and set the sound in the individual games (mostly within each games setsound or setup utility).

Than about the Mouse, does it have a switch underneath saying PC and MS, or something like that? Just try the other setting. Normally when you start DOS (I assume you have dos installed) and the Mouse driver is loaded (Mouse.com or CTMouse.com) that is everything needed. What kind of mouse is it?

Does sound like something is really not happy if the HDD interferes with video.
Where do these glitches show? from the command line or only in games?
Pull as much hardware out as possible load the PC bypassing startup files and see how it behaves as a 1st step.

The only hardware connected to the motherboard is the I/O and graphics cards right now.
The mouse is a 90s Belkin PS/2 mouse with a serial adapter. The computer came with an old serial mouse that behaved the same way, and so I tried replacing it but the mouse itself wasn't the issue.

I'm not 100% sure the graphical glitches are being caused by the hard drive. However they do usually occur when the HDD is being accessed. It doesn't happen from command line or text-only programs, just graphical video modes. It will sometimes flash horizontal lines or distortions across the screen and then there will be partial horizontal lines of wrong-colored pixels. In some games, it actually affects gameplay. I've had it create pixels that can be hit by projectiles in Scorched Earth. If I had another good video card I would test that too but at the moment I don't.

Vibra 16c just has these jumpers. I don't think they affect IRQ because it's supposed to be a plug and play card.

EDIT: I plugged the video card into a different slot. It doesn't glitch as much, and not when the hard drive is active. Maybe I just have a crummy video card and I/O card. Or a crummy motherboard. I also managed to get it to detect the hard drive when I plugged the sound card back in. I don't know if the sound card is enabled though. I'll test that again tomorrow.

Moved the sound card, ran CTCM, got sound!
I suppose I should just look into getting replacement I/O and graphics cards. For the graphics I'm looking somewhere on the cheap side. I'm not planning to play anything more taxing than X-Wing.

I have no idea what kind of I/O card to look for. Does it matter very much what I replace it with? Just something that will work with a 486.