The power from USB and FireWire use different voltage, so that questionable cable may have damaged some circuitry on the logic board, although the part that handles battery charging still works. And I think you're saying that when it's disconnected, it still works as an iPod (plays songs).
You've done the usual trouble-shooting steps. If you have a different FireWire cable, try it to rule out bad FireWire cable as the cause. Also, if you have another FireWire device (like an external drive), connect it to confirm the computer's port is working normally.
I would let it play songs unattended (disconnected) with headphones at normal volume (or higher), until it runs out of battery power. Then, keep trying to turn it on until nothing appears on screen anymore. Basically, make the battery completely dead. NOTE: You could open the case and disconnect the battery, but the 1st and 2nd gen iPods are harder to open and more prone to getting damaged, even if you use the proper tools. But if you know how and you've done it before, you can do that instead to save time.
Then, connect the iPod to see if there is any difference.