The said covenant (ahd) is correct and one should act upon it if the profession is required to prevent harm from reaching Muslim women. It is not obligatory to act on it if there are enough Muslim female physicians. It is permissible to take up such a profession, and it is permissible to deliver babies when necessary.
It has been reported in some narrations to perform the Istikhara with the actual Quran just before offering the prayer. What you mention does not relate to this.
Usually after washing the eyes there is nothing left of the kohl except the colour, and there is no trace of it that prevents the water from reaching the skin. Therefore, the ghusl in this case is valid.
