How do I know when my on-board ethernet is dead? (Realtek 8112L LAN controller w/ AI NET2 )

Try reseting the CMOS by removing the battery or jumper method (moving the jumper from pins 1-2 to pins 2-3). Did you lose power recently or brown outs? If your onboard NIC is shot, you could always purchase a PCI NIC card for cheap.

You've pretty much done everything except one easy check: the easiest check with a known working connection is to plug into that exact cable and look at the link lights on the ethernet port; if the card is completely shot they'll probably not be on whatsoever. (This isn't foolproof but if an ethernet port gets shorted out from static this will almost ALWAYS result. It's certainly the easiest first thing to check).