france-eesti is almost right (nice job!
)
For a single hair it's "hajszál", hair strand.
Találtam egy hajszálat a földön. (I found a strand of hair on the floor)
Találtam egy hajszáladat a földön. (I found a strand of your hair on the floor) (However I don't see myself using this sentence ever)
With "szőr" it's "szőrszál".
'he has hair on his legs' would be "szőrös a lába" (his legs are hairy) you can say "szőrszálak vannak a lábán" (he has hair on his legs) it's correct grammatically but not used (maybe in specific case)
For teeth you don't need to specify (you can, just don't have to) it's about one tooth.
A lányomnak fáj a foga. (My daughter has tooth pain)
A lányomnak fáj az egyik foga. (My daughter has tooth pain in one of her teeth)
Both sentences are correct, the first one is maybe used a bit more, but both is used. If all / most / more of her teeth would hurt it would be:
A lányomnak fájnak a fogai. (My daughter has tooth pain in more of her teeth)
A lányomnak több foga is fáj. (My daughter has tooth pain in more of her teeth) - it's a bit more natural way to emphasize it's about more teeth
A lányomnak fáj az összes foga. (My daughter has tooth pain in all of her teeth)
Glottomany you're right "haj" is on the head, "szőr" is everywhere else and on animals.
A macska szőrös. (The cat is hairy)
Eyebrow is "szemöldök" and "szemöldökszál" is a strand of your eyebrow.
Beard is "szakáll" and a strand of it is also "szál" (szakállszál), but rarely used like that. If you leave your hair from shaving your beard everywhere it's referred as "szőr" or "szakáll".
Tele van a csap a szakálladdal. (The sink is full of your beard)
Tele van a csap a szőröddel. (The sink is full of your hair) - not knowing /specifying what kinf of hair
Otthagytad borotválkozás után a szőrt a csapban. (You left your hair in the sink after shaving)
'she had her hair cut' is "levágatta a haját".