While the men of the house are away, can women open the Perumal Koilazhwar (the box in which Perumal is kept) to offer prasadam? Or must they offer food mentally without opening the box?

It is not in custom for women to touch the box kept for Perumal.
Women do not touch the box even though the box is made of wood and Perumal is kept inside the box; it is not known what impurity might befall when a woman touches the box; as it is revered as Koilazhwar with a separate sanctum, it is not a practice for women to open Koil azhwar.