The hot towel is an old barbershop pampering tradition. The idea is to soften the whiskers as they soak up water, while the barber does something else. Too hot water is not good for the skin and can lead to dryness, though. You can accomplish much the same thing by just lathering up and leaving the lather on the face for a few minutes, then rinsing off and lathering up again the shave.
Menthol cooling is just a sensation from the nerves in the skin. It should not affect the shave result.
Alcohol is a good cleaner, solvent and disinfectant. The idea is to prevent infection and some aftershaves also add humectant ingredients and fragrance. Most soap has a higher pH than skin, rinsing with water and applying some post-shave treatment is supposed to restore normal skin pH. Some aftershaves have lactic acid or similar ingredients which can provide some skin toning effects.
I don't shower before I shave or use a hot towel. After spashing water on and rubbing it in to get water all around the whiskers, I start loading the brush. With the first little bit off soap I paint it on and add a little water. That's my pre-shave.
By the time I get the brush fully loaded and face lather I'm good to go.
I do use a pre-shave cream (Proraso), when pressed for time, or traveling where my chances at more than a gallon of water or hot water are limited.
YMMV of course.
What have they been up to in Rivendell since they perfected Arko?