Why would hydrogen not be generated at the cathode when electrolysing zinc sulfate?
In an electrolytic cell we are using the electrical energy from an outside power source to drive a chemical system against the direction it would spontaneously move.  In this case, we can take a solution of zinc ions and force them to become zinc metal.  As long as we put the zinc ions in an environment where they can be reduced, they will be.

Consider the half reaction that occurs when zinc metal is produced:
$\ce{Zn^{2+}(aq) + 2e- -&gt; Zn(s)}$

This is a reduction reaction, and reductions always occur at the cathode, whether the cell is galvanic or electrolytic.  

The other half reaction involves hydroxide being converted into water and oxygen.
$\ce{4 OH- -&gt; 2 H2O + O2 + 4e-}$

This is an oxidation reaction, and oxidations always occur at the anode.

Platinum and carbon are nonreactive in this system, so they can be used for either electrode.  Zinc can be used as a cathode because that's where it is being deposited, so we're just adding additional metallic zinc onto our already existing zinc electrode.

With the idea that oxidation occurs at the anode in mind, what would happen if we used a zinc anode?