I have an very old Knoppix disk. I tried booting from it and got this:
Can't find KNOPPIX filesystem, sorry. Dropping you to a (ver limited) shell.
Press reset button to quit.
Additional builtin commands available:
cat mount umount insmod rmmod lsmod

Anyway, I thought I'd give it a try.

At this point, I might just try installing a different version of Linux.

Thanks again for your help.

I downloaded antiX. When booting from the disk, it would not give me the GUI unless I chose the equivalent of "safe mode."
I am installing it, anyway, just to see what happens, but I don't have high hopes. I don't know why the BIOS would be the issue, but I'll try to get that updated. I think there is one newer version.

Well, antiX DOES boot to a GUI! However, I don't have any video option better than 1024x768, and I don't know how to change that. I did try loading NVIDIA drivers. After that it would boot as though it wanted to use both screens, but I could not get a GUI. I did ctrl+alt+F1 (or maybe F2) to get to a prompt and ran startx (by itself, just "startx"), and did not get a GUI. As I recall, though, I did get a mouse pointer that would travel from screen to screen.

I also updated the BIOS and reloaded Mint MATE with no improvement in my black screen issue.