Is it strictly necessary for a married woman to keep her hair covered in the following situations? If not, how important is it?
1) When alone in the house
2) When alone (or with husband) in the bedroom
3) When chinuch-age children are present
Magen Avaraham Orach Chaim 75:2 says that in her own house, a lady does not have to cover her hair, however the Bais Shmuel in Even HaEzer 115:9 and the Bach says she does have to cover her hair. In Igros Moshe Even HoEzer 1:58, HaRav Moshe ZATZAL says that the Bach is talking about in a courtyard; however, inside the home, it is permitted.
As she can uncover it her house, then it would definitely be permitted in the bedroom alone with her husband.