Effect of pH on electrolysis
I'd rather not put it in such terms. Dissociation of water is determined by the constant $\ce{K_w=[H+][OH-]=10^{-14}}$, which you can't change simply by adding acid (or anything else, for that matter). Acids dissociate easily into ions, just like you said, therefore adding acid to pure water (which is a poor conductor by itself) makes the latter more conductive and thus more prone to electrolysis.

Electrolytic refining of copper is a whole different story. Indeed, once dissolved in water, $\ce{CuSO4}$ would dissociate into ions completely, no matter whether or not you acidify it. The role of acid is just to prevent hydrolysis.

In general, as you change pH, some of your ions may undergo certain transformations (like $\ce{CO3^2- -&gt; HCO3- -&gt; CO2\uparrow}$, or $\ce{Fe^2+ -&gt; Fe(OH)2\downarrow}$), which in turn might or might not affect the reactions on one or both electrodes. As it happens, being too general prevents us from drawing more specific conclusions.