All, update - sorry for long delay, life happens. I did execute the firmware update procedure from the URL listed up-thread. It went seamlessly (I held the programmer’s button in with a deer antler from the greenbelt behind my house - is that a unique way to update an iMac?) No change in the ethernet behavior. I did notice that when I command (via System Preferences -> Network (ethernet) -> advanced -> hardware (I think, not on the 10.4 machine right now) the rate to 100TX full-duplex, a) the “connect” light on the hub will turn on, and b) the “power/busy” light will go off for a while - presumably indicating full blast messages swamping the hub for a few minutes. I looked at the Service manual .pdf for that model, and the battery holder is indeed close to the ethernet port (the whole I/O subpanel, in fact).
Next stop, I’ll pull the cover off the machine and see whether I can spot any spilled battery electrolyte from the old battery or anything else suspicious looking on the logic board in that area, and will report what I find. - Mark