All other primates are liberally covered in thick hair, so I think the real question is why we lost thick hair in other locations but kept it in the areas you mention. One possible answer is that humans needed to be better able to cool themselves in the act of chasing game on the African savanna. See Modern Human Origins.
So why keep it in the under arms, the pubic area, and on the face and head? I think most of the answers are anthropological rather than medical. Since I am a doctor, I am going to offer you guesses. First guess: the under arms and the pubic region hair may be a need to prevent friction at locations where body folds would be otherwise rubbing against each other when walking or running. The prevention of skin against skin contact may also aid in keeping those skin folds drying by allowing air to circulate betted. The areas are not very dry, but some air circulation would prevent the skin form staying moist all the time. When skin stays moist all the time it tends to get yeast infections. Second guess: facial hair in males signaled sexual maturity and this was "useful" in some sort of social purpose. Third guess: Eyebrows accentuate ability to see facial expression and may have helped in communication in social situations. Fourth guess: Head hair keeps sun off the scalp and would prevent solar injury.
Voice change is driven by male hormones and generally occurs around age 15. The size of the larynx increases more in males than in females and the vocal cords thicken more, too. Larger resonant cavities give lower notes just as larger musical instruments give lower notes. Why this should be so from an evolutionary perspective is perhaps less clear. Low notes travel better in air so perhaps there was an advantage in communication over distance among hunting bands.
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