As you know, for sometime the Latin Rite has utilized female lectors, ushers, altar servers and Eucharistic Ministers. What, if any, functions are open to women during Eastern Rite services?
In the Eastern Catholic Churches women can (and do) serve as cantors, lectors, and ushers/greeters. Some of our very best cantors are women, and in the Eastern Churches cantors play a crucial role. It is difficult to celebrate the liturgy properly without a trained and experienced cantor.
Is a true that a woman cannot approach the sanctuary except during church cleaning?
In the Byzantine tradition women are generally not allowed to go behind the icon screen (there are exceptions). However, this isn’t a judgement against women or anything of that nature. It is simply that the sanctuary (which we call the altar) is a sacred area, and no one is allowed in there unless they have a specific reason to be there. I, as a layman, am forbidden to enter there unless I have been told to do so. Even the priest only enters the sanctuary when he is carrying out a liturgical function, or is making preparations to do so. Since in our tradition we do not have altar girls, there usually isn’t a reason for a woman to enter into the altar.