We have to get the principles right before we can know what's appropriate in various cases. The practical principle is male headship, which means it's always better if men take the lead. When both men and women understand and accept this, knowing what's appropriate in all the various cases isn't much of a problem. They will both sense when the headship of the man is or is not being respected, and they will adjust their behavior accordingly.
For example, a wise woman directing a choir will take special care not to lord it over the men in the choir. She will not order them about or correct them curtly, as she might women or children. She should defer to them on some matters, for instance, asking them to lead any prayers said before practice or before meals, and asking them to chant the parts of clergy if needed. I have know several female choir directors, typically older women, who were very good at treating men respectfully, but I have also known several who treated men like women and even like children, which is especially wrong.
It would be better if choir directors were male, since choir directors do plainly exercise authority over their choristers, telling them what to do, where to stand, when to sing, how to sing, etc. The centurion tells us what authority means:
"For I am a man under authority, having soldiers under me: and I say to this man, Go, and he goeth; and to another, Come, and he cometh; and to my servant, Do this, and he doeth it." (Matt. 8:9)
It is not natural for women to relate to men this way, and we should avoid it whenever possible.
Teaching is often also an exercise of authority. The Romans addressed their teachers as magister
— from which we have master
. The Apostle Paul speaks of teaching and authority in the same breath; the Fathers do as well. They allow that women might have something to teach men and might do so informally, through conversation (as Aquila and Priscilla taught Apollos), but they explicitly and consistently forbid public teaching of men by women. The sources to cite are many: St. Cyrian of Carthage's precept #46, St. Basil the Great's rule #73, St. Jerome's letter #127, St. John Chrysostom in many places, Canon 70 of the Sixth EC, the Didascalia, the Apostolic Constitutions, among others. Reading the Gospels and Epistles aloud in church was also considered teaching and was reserved for the priestly offices and forbidden to women. (St. Jerome faults Pelagius on this point.)
Of course women can teach other women as well as children, but it is better if men teach older children, especially older boys, who are trying to become men and greatly resent being forever mothered. Boys aspire to be men not women, so we should give them men as teachers and leaders. We should also teach boys to be leaders and not make them subject to their elder sisters if the age difference is slight. St. John Chrysostom actually says that sex trumps age when it comes to precedence, "since among the children the female doth not possess equal sway." (Homily 34 on 1 Cor.) No one, of course, would put a boy of six in charge of a girl of 14, but it would teach boys and girls something important if the boy of six were asked to say grace before supper when his father isn't there.
The Fathers do teach that greater modesty is expected of women. Some ancient Christians objected to women writing books. Nowadays everybody has become so immodest that that hardly seems an issue, but women should give more consideration to how forward they are being. God does give the lead to men; it is therefore only fitting that the Church's leading writers and teachers be men. If instead they are women, something's wrong. It may be that men are weak in faith and not living up to their responsibilities, but it may also be that both men and women are weak in faith and not living up to the Church's teaching with respect to the sexes.
The Church's public worship should reflect the order of creation so that men and women both understand, and feel called toward, their respective responsibilities. Men should therefore be easily seen as in charge. What then is the reason to create offices for women or girls, making them responsible for passing the collection basket, controlling entry into the church or the flow toward the Chalice, or holding the cloth at the Chalice? Are such things not often deliberately intended to equalize the roles of men and women, out of the mistaken view that the Church is too male-dominated? Sometimes the justification is that this is a way to interest girls in church, but I have raised two faithful Orthodox females to adulthood, and neither ever complained about having nothing to do in church. I believe that complaint originates with the parents, who have wrongly taught their daughters that it's unfair for girls not to be allowed to do everything boys do.
Finally, it only makes sense, if we truly believe what the Church has always taught about men and women, that our councils would be composed of men. It's their responsibility. It's important that they believe that and feel inspired, by the honor shown their manhood, to step forward and assume responsibility for the community. The welfare of the community depends on men doing so, for when men are not taking responsibility for the community, women and children suffer greatly.
I think that about covers it. Let me know if I've missed anything.