It is not permissible for a menstruating woman to stay in the mosque. As for passing through the mosque, there is nothing wrong with that. Thus, it is not permissible for her to go there to listen to Halaqahs (study circles) and recitation of Quran, unless there is a place outside the mosque where the sound can reach via loudspeakers, in which case she can sit there and listen to the Dhikr.

A woman does not have to undo her hair when she does Ghusl following Janabah (impurity following sexual discharge), but she must make sure that the water reaches all parts of her body, including the hair and its roots.