I decided to haul the machine out of the closet and give it another go. It occurred to me that I should probably update the BIOS on the computer first. It might improve the compatibility. It turns out that the machine has BIOS rev A03. Dell's last release is A15!
I installed the latest BIOS. The end result is that a lot more is functional now. For anybody having trouble installing to a machine that (on paper) should meet the requirements, I recommend that you see if you've got the latest BIOS installed. (BTW, apparently the BIOS upgrade installers that run under Windows are notoriously unreliable. Many folks claim that they risk bricking the computer. You should instead use the DOS installer. It's a royal pain, but the safer route.)
So anyway, I did the Qubes install. After a few hiccups, I got the thing installed. It has been running fine for a whole day now, so attached is the HCL report.