Is the turnover # dependent on [E]? According to the formula, K3 = Vmax/[E] it looks like it is. But on one of the enzyme kinetics handouts, k3 is defined as the maximum # of molecules of substrate converted per molecule of enzyme per second. This definition implies, to me, that the turnover # is a quality independent of enzyme concentration.

You are correct in thinking that the turnover # is a quality independent of enzyme concentration. The formula k3 = Vmax/Eo simply represents a way to calculate that quality, given data you can collect (measure Vmax, measure the amount of E that gave you that Vmax).