To keep this concise..
The base position amongst the madhabs appears to be that a woman cannot talk to a ghayr mahram without need.
Does performing dawah/conveying Islamic education constitute need.
For example, a sister preaching the good to her male relatives. Or a sister teaching a male student via direct contact.
If the answer is no, how did the great female scholars of years gone by convey ijazas upon their male students and learn themselves. Always in the presence of a mahram?