Does a deserted mosque, designated for teaching women the Quran, retain the ruling of a mosque, and is it permissible for a menstruating woman to enter it to seek knowledge?

As long as the old mosque remains in its original state and is suitable for prayer, it holds the rulings of a mosque. Women's prayers, their recitation of the Quran, and their learning are permissible within it, except that a menstruating woman is prohibited from staying inside.