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.

It is not Sunnah to recite Qunut in Fajr prayer all the time. Your prayer behind an Imam who recites Qunut in Fajr is valid, but if there is someone who does not always recite Qunut in Fajr, it is better to pray behind him, so as to follow the Sunnah.

Two people are enough to pray in congregation, whether that is in the house or elsewhere. However, men are required to offer the prayers in congregation in the mosque. It is not permissible to offer the obligatory prayers at home, whether in congregation or alone, unless one has an excuse.