“1) There is no schita regarding hair. The problem would be with the towel. People use the mikve on Shabbos or Yom Tov – which gets hair just as wet as a shower.”
Squeak, I don’t know how you wash your hair, but when I do, there is DEFINITE sechita. How can you possibly shampoo and rinse hair without some form of sechita, unless your hair is SO short that there really is nothing to squeeze? And yes, the towel is a problem, yet strangely, we wash and dry our hands for Ha-Motzi on Shabbos, and women who go to the mikvah on Friday night are allowed to gently pat the excess water from their hair with a towel, without squeezing it. If you are not using shampoo, there is really no reason for the head to get wet altogether when showering. An adult can easily shower without the water going on the head. And if not, then it is perhaps time to invest in a hand-held shower attachment so that only the soiled parts of the body are washed.