By holding copper hand rods (or applying voltage to any part of the body) wouldn’t the zapper cause electrolysis which in itself could cause harmful by products that damage cells?

It will cause electrolysis which will cause some free radicals and some cell damage. But we use low voltage and pulsed current so it is not really much of a problem, as general research about body electricity shows. But high voltage with direct current applications would definitely be a problem, as they can destroy red blood cells.