In the ongoing fight to get my desktop system fully functional again, I feel I've pretty much eliminated monitor problems from the equation. I just finished two quick tests. I moved the monitor to my wife's machine and hooked it up via the DVI input. No problem. I then moved it back to my machine, pulled out the new graphics card, and ran the motherboard's on-board VGA output to the monitor's VGA input. No problem. In fact, I'm using that VGA hookup now, which is the first time I've been able to use the desktop for the past several days.
All signs at this point are aiming squarely at the motherboard, and specifically the PCI Express slot in which the video card sits. The rest of the board seems to be working fine, as my presence here (hopefully) indicates. While it's nice to have a fairly solid target at which to direct my next efforts, it's not the target I wanted. If it were the monitor or video card, it wouldn't be much of an issue. Get a new one, plug it in. With the video card, there would be drivers to install, but that's still a piece of cake compared to the work involved when swapping motherboards.
A new board typically means reinstalling Windows from scratch. At least that's what it used to mean. However, I just found a couple web pages that have detailed instructions on how to swap motherboards without affecting Windows. It's still more work than installing a new video card or monitor, but much less work than a full re-install.
So....here we go!