Does the convergence rate of multigrid depend on the total number of smoothing steps or on the number of pre and post smoothing steps seperately?
Separately, but it does depend. Not very strongly, however: A very large number of pre- and post-smoothing steps only improves the convergence rate a little bit over a large number of steps. The difference is most between using one, two, or three pre- and post-smoothing steps.
And you need both pre- and post-smoothing steps. You cannot compensate for no pre-smoothing steps by using more post-smoothing steps.