After much efforts, trial-and-error, troubleshooting, and research, I hereby conclude that the networking does not work on this desktop. While the old/ancient WiFi card was an addition to the box, even the wired LAN ports would not work.
The WiFi card just wasn't ever detected. This didn't entirely surprise me since the card is SO old, but it was slightly disappointing. There seems to be some possibilities on getting this to work with some online updates of sorts, which brings me to the wired LAN port...
The LAN port, detected a link (or lack thereof), but would never actually complete a DHCP handshake (though a 'DHCP Discover' was sent, and a 'DHCP Offer' was sent back, it never seemed to process beyond that). When configured with static IPs and settings, it never showed up in any ifconfig commands, did not reply to ARPs, and just didn't seem to work.
This same desktop (including ancient WiFi card) work just fine booting a live Fedora USB (and Kali and a few others too).