Is body hair a functionless vestige?

The problem seems to be that you think vestigial means functionless. It doesn't. Vestigial means having a secondary or rudimentary function compared to the same feature in another species. The sensory function of hair on humans is a secondary or rudimentary function compared to the primary function of thermal regulation. Hair is vestigial on humans. We depend on sweat and clothing for thermal regulation.

This is a secondary role for hair, not it's primary function which is insulation. This is why human hair is considered to be vestigial because it has lost its primary function. Hair also provides cover for ticks. When someone is bitten by a tick they often find them just inside the hair margin. Also, body lice are evolved so that the distance between their pincers is the same as the distance between human pubic hair. The best cure is just to get rid of the hair.

You should take a trip to Africa sometime. You will find that keeping the body cool is much more important, especially to hunters. Humans evolved a very interesting way of hunting. We chase animals during the heat of the day until they collapse from heat exhaustion. We don't collapse because we sweat profusely. Cooling the body is a much more important adapation for humans than what little warmth our hair provides.

No, it doesn't. If anything, it keeps more water on the body than not having hair. For examples of hair that DOES keep water off the body you should check out dog breeds that have a dense undercoat, such as Labradors or Newfies. That hair actually does keep water off the body unlike ours. You seem to see a lot of things for which there is no evidence. Hair density in women is the same as in men for most cases. Alopecia is more common in human men, but it does occur in women as well. Also, more insulation is actually a hinderance in many climates that humans are otherwise well adapted to, such as the savannas of Africa. Like I said before, sweating is a much more useful thermoregulatory adaptation in humans than hair is, by a long ways.