If I use denatured alcohol right away to clean up, even before the epoxy starts curing/hardening, does that in anyway retard or prevent the curing of the epoxy?

Using solvents like denatured alcohol for cleanup is established practice in boatshops. I've used a gloved finger dipped in alcohol to smooth "gelled" epoxy fillets on dozens of boats and there have been no problems. No bad cures, blush, nothing. If the solvent penetrated deeply into the curing epoxy it would probably disrupt the cure. But in all instances where we advocate the system we warn builders to wait until the epoxy has gelled before smoothing fillets.

Other solvents work just as well but denatured alcohol is somewhat safer. In any case, be careful to ventilate your shop whenever you work with solvents.