Is it permissible for a woman to deliver sermons and lessons in the mosque via loudspeaker, knowing that her voice can be heard by men while she is in the women's prayer area?

The voice of a woman, in itself, is not awrah (something that must be concealed), as Allah Almighty says: "And when you ask [his wives] for something, ask them from behind a partition" and "Then do not be soft in speech [to men], lest he in whose heart is disease should covet, but speak with appropriate words." The prohibition against a woman being soft in speech and making her voice alluring is out of fear of fitnah (temptation) and the pleasure of the listeners. The most correct view is that a woman should not give lessons and sermons via loudspeakers, out of fear of men deriving pleasure from it, and she should limit her voice to women and her mahrams (unmarriageable kin).