1- Does applying kohl invalidate the fast of a woman, whether it is an obligatory or a supererogatory fast? 2- Will a man be rewarded if he grants his wife permission to observe a supererogatory fast?

Applying kohl does not invalidate the fast unless its taste or color reaches the throat. Even if it does, the fast is still not invalidated according to the sound opinion, because it is not in the meaning of eating and drinking.

It is not permissible for a woman to observe a voluntary fast while her husband is present except with his permission. This prohibition is due to the affirmation of the husband's right over her. If he permits her, he will have the reward for obeying Allah's command and the reward for assisting in good deeds.