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.

If a woman in nifas sees that she has become pure before 40 days are over, then she should do ghusl and pray and fast, and her husband may have intercourse with her.