I've set up a Windows XP Professional SP3 VM in Bochs, which with I can change what CPU is being emulated. I'm not going to embark or anything, because any performance metrics from that are a product of the speed of the host computer, not the CPU being emulated. I'll be using a weaker criteria for "running": running the Object Testing Arena. To help identification and as a blatant "this is not real hardware" sign (look at the clock speeds), I've also put a CPU-Z window in the screenshots. The version being used is 47.04, Win32 SDL.
(https://i.imgur.com/mNkxDHD.png)
(https://i.imgur.com/63DtCXC.png)
(https://i.imgur.com/9fGWNQp.png)
(https://i.imgur.com/27GXqhV.png)
(https://i.imgur.com/vUkDOBs.png)
(https://i.imgur.com/9ORCF5F.png)
It's a foregone conclusion that anything older will not work. From this testing, it appears that SSE2 is a requirement for 47.04, not just SSE. This makes the Willamette Pentium 4s the oldest CPUs capable of running DF (at least for Windows), assuming the emulation is accurate. If anyone has real hardware and is willing to try and confirm these findings, please do.
(That website I linked back in the OP was for an older version, judging from the publication date. I cannot test the legacy Win32 version because it requires OpenGL, and Bochs doesn't seem to emulate an OpenGL device (or I haven't checked around hard enough). It throws an error instead, stating that DF "is not a valid Win32 application". It does work on the host system (Windows 10 x64), so I think Windows XP is being drunk.)
I have spent days trying to get DF to work on Linux. Not because of DF being annoying, but because the virtual machine(s) I've been using are horrendously slow. Same stuff about a weaker definition of "run" (title screen and Object Testing Arena only) and not testing performance applies here. You'll have to take my word for the CPUs; I couldn't be bothered to get a CPU identification utility (something like CPU-Z), because this thing is unbearably slow. I've included a screenshot of the setup in Qemu Manager (yes, it's very out of date, but it's the only one that worked) for verification.
(https://i.imgur.com/HUYIPyx.png)
(https://i.imgur.com/2fnhDST.png)
(https://i.imgur.com/BO7OpCm.png)
(https://i.imgur.com/XWZySue.png)
(https://i.imgur.com/yRVWtBk.png)
In other words, 32-bit Linux DF requires SSE, but not SSE2. It would be pointless, but it's impossible to test anything older; Puppy Linux (this one is based on Ubuntu 19.04) will not boot on a Pentium due to lack of features, specifically PAE and CMOV. Qemu (the ancient version that I'm using, at least) doesn't have the option to emulate a Pentium Pro, so I can't do anything there.