increasing voltage can allow different chemicals to react, but I assume that the strongest oxidant/reductant pair will always react, and these have the smallest Eo difference [with the exception of chlorine, which reacts in preference of water in higher concentration, and i believe that also changes its Eo value].
and as for the other question, I'd think that constant resistance cannot be assumed. the energy required for the reaction to occur is is the Eo difference, and it may very well be that electrons from the supply can have higher energy, this energy would be converted to kinetic energy in the reaction, i.e. the solutions/electrode heats up slightly. Hence the more energetic the electrons are [voltage], the more heat is released [current].
Another way to combat this simply is to change the power output of the supply, though I really don't think it's of any relevance to the study design =]