I'm going to jump around a bit..
If either is used ON camera, can it trigger the other?
Yes, but only optically. The 580EX-II can be a wireless master on a camera (or on a TTL Cord). The same is also true, but if you're trying to trigger non-RT flashes (anything but a 600EX-RT), then you have to do it optically.
Am I correct in thinking that the 5DIII needs a cord or separate trigger unit to trigger a 580EXII, but it will (via radio signal) trigger the 600EX-RT?
This is a confusing question, but here's your answer -- to trigger any flash off of the 5D3, you need either a cable connected to the flash off the camera, or some sort of trigger -- that trigger can be Canon's ST-E3-RT Radio Trigger for 600EX-RTs, ST-E2 trigger for optical trigger (which will trigger any optical slave), or a third party trigger (Such as PocketWizard Plus system or TT1/TT5s, Pixel Kings, etc).
To clarify: the 5D3 will not, without a trigger/transmitter, fire the 600EX-RT via radio. It doesn't have the capability built in, and to my knowledge, no camera will have radio firing built in, as this would kill future sales of the ST-E3-RT radio transmitter.
If I use both a 600EX-RT and a 580EXII off camera, can I use the 600 to trigger the 580?
I think I addressed this earlier for you; basically, yes, but only optically. You need one to be the master and one to be the slave; the 580EX-II only has the one slave mode (optically), and the 600EX-RT has two master modes (radio and optical).