"I washed my hair in the bath."
On a cultural note, be aware that in Japan you are generally expected to wash your hair and body outside of the bath, rinse, and then enter the bath for soaking. It is common for the same bath water to be used by multiple family members (or patrons at a public bath) so it is considered bad to make the water soapy by washing yourself in it.
I'm curious if they intended the sentence to be something you could say in a conversation about cultural differences if you come from a culture that washes their hair in the tub
Men usually use ふろ whereas women usually use おふろ. If you are talking to someone you should speak to politely, you use おふろ whatever your gender is.
The particle に can be used when talking about places you went to: うみ に 行きました。I went to the beach. Whereas で is used when talking about places where you DID something: うみ で およぎました。I swam at the beach.
に is an indicator of a destination or a place of existence. 学校に行きます = I go to the school. 学校にいます = I am at the school.
で is the place of an action レストランで食べます = i eat at the restaurant.
In many sentences both can be used, but the meaning becomes different. そこに投げる = I throw it to there. そこで投げる = I throw it from there.
No, when describing location, rather than motion, に and で are never interchangeable. Which one to use depends on the verb.
Why is it not 髪の毛 instead of 髪？ is there there different contexts that one is better than the other.