I'm an idiot, guys. I assumed I was on the latest bios but I guess you can't take such things for granted with such an old motherboard. Updating to the most recent (which is still from over 10 years ago) resolved all of my issues. I can run the stable release of 12.02 (non-legacy) and get video and audio just fine over HDMI now.
No idea why it made a difference as the notes for this bios just mentioned fixing USB3 issues, but yeah. Sorry to have wasted everyone's time.