I sorta go back and forth between wash solution (often a stronger-than-normal mix in a spray bottle) and dedicated clay lube. Much as I hate to admit it, I believe that the clay lube DOES provide a bit more lubrication. I don't think this usually matters, and I still clay while washing, but if I had a black car I'd probably use the lube.
I do rewash the panel after claying, though as BillNorth said, you really shouldn't have to if you clay correctly. But it gives me peace of mind; I don't like the idea of leaving clay lube on there like a QD. When I use car wash soap it only takes a second to rewash before rinsing and I worry about the soap breaking down the clay and thus leaving clay residue on the panel.
Yeah, car wash solution DOES break down some clays. Griot's, for instance, practically dissolves in car wash soap. The Autopia Block, on the other hand, doesn't. Different types of clay call for different procedures.