JazakAllah for your question. Firstly, regarding female teachers and the issue of menstruation. This question has been previously answered:
Secondly, regarding women not allowed to enter the Masjid whilst in the state of menstruation. The area of the Masjid being referred here is the prayer area. The below in sha Allah is a more detailed answer:
“If the masjid is a permanent masjid that has been dedicated by the community for such and is considered a permanent endowment or like an endowment. This means that is not a rented apartment, house or industrial unit which is used temporarily until the community can afford to purchase a permanent place.
In the above case it would be permissible for menstruating women to enter into the prayer area and take part in classes. However, since it is used as a masjid for the daily prayers then menstruating women should not enter into the prayer area unnecessarily. It would be superior to arrange seating for all the women or at least the menstruating ones in an adjacent room. However, if this is not possible then menstruating women could also enter into the women’s prayer area of the prayer hall.
In the case that it is a proper masjid then it would not be permissible for menstruating women to enter into the prayer area [the shar’i masjid] for this or other reasons. However, if there are multi-purpose rooms within the building which are not designated permanently for prayer (as is found in many Islamic centers nowadays), then menstruating women are allowed to enter into those rooms for classes. Basically, a masjid is the room or hall that has been permanently designated for prayer, even if there are other rooms in the building.
And Allah knows best