yes, the 600 and 580 II will working together, but only in optical mode.
I have sold my st-e2 and two 580's and upgraded to st-e3-rt and two 600.
Sofar, I have been able to set everything that I needed with the camera menus. For example, set to HSS on menu and slaves will switch to HSS. With the "old" 580, I had to do an additional step: set HSS on the flash as well.
The on flash menus are so MUCH easier to use on the 600.
It was a bucket of money, but it sure is worth it.
I will need to spend more time to explore all the additional functionality, which is a lot.
Most important for me is the radio triggering, no more need for line of sight, which really helps when you use a small softbox on the flash