What is a woman's role in the church? What can she do and/or not do?
Note: I'm not sure this makes any difference, but this question was asked by a man.
Yours is a tough question, not because of what the Bible says but because of people's reactions to it. Anything I say in comment about the passages below will probably bring somebody's disagreement down on me.
There are really only two or three passages that deal with the woman's role in the assembly of the church.
"I will therefore that men pray every where, lifting up holy hands, without wrath and doubting. In like manner also, that women adorn themselves in modest apparel, with shamefacedness and sobriety; not with broided hair, or gold, or pearls, or costly array; But (which becometh women professing godliness) with good works. Let the woman learn in silence with all subjection. But I suffer not a woman to teach, nor to usurp authority over the man, but to be in silence. For Adam was first formed, then Eve. And Adam was not deceived, but the woman being deceived was in the transgression. Notwithstanding she shall be saved in childbearing, if they continue in faith and charity and holiness with sobriety." (1 Tim 2:8-15)(That this may be in the context of the assembly is shown in verse 8. It should also be noted that Jewish women were exempted from praying or attending the assembly, or any religious duty that was to be performed at a set time because of the responsibilities of child rearing; exempted, but not forbidden to assemble or to pray.)
That passage particularly and specifically prohibits a woman from taking authority over a man. That would disqualify her from the eldership. The verse also leads to a touchy situation in the churches of Christ. Because we believe that the subjects of baptism are those who are of an age to understand their sin, it's consequences, and God's grace, women usually are permitted to teach the Bible classes for the younger students. However, all of the congregations I have attended regularly have determined that men will teach classes which may have baptized boys (generally fourth or fifth grade and above). This is to avoid any possibility of violating this verse by having a woman teach a "man."
"But I would have you know, that the head of every man is Christ; and the head of the woman is the man; and the head of Christ is God. Every man praying or prophesying, having his head covered, dishonoureth his head. But every woman that prayeth or prophesieth with her head uncovered dishonoureth her head: for that is even all one as if she were shaven. For if the woman be not covered, let her also be shorn: but if it be a shame for a woman to be shorn or shaven, let her be covered. For a man indeed ought not to cover his head, forasmuch as he is the image and glory of God: but the woman is the glory of the man. For the man is not of the woman; but the woman of the man. Neither was the man created for the woman; but the woman for the man. For this cause ought the woman to have power on her head because of the angels. Nevertheless neither is the man without the woman, neither the woman without the man, in the Lord. For as the woman is of the man, even so is the man also by the woman; but all things of God. Judge in yourselves: is it comely that a woman pray unto God uncovered? Doth not even nature itself teach you, that, if a man have long hair, it is a shame unto him? But if a woman have long hair, it is a glory to her: for her hair is given her for a covering. But if any man seem to be contentious, we have no such custom, neither the churches of God." (1 Cor 11:3-16)
This passage would seem to indicate that women may have been allowed to participate in the assembly, particularly if they had miraculous gifts. However, even in this passage it is clear that a woman was subject to the man in the assembly.
Romans 16:1, particularly in modern translation, may apply to a woman's role in the church as well. It speaks of one Phoebe as a servant or deaconess. That this is not the same as the position of deacon is probable, because she obviously could not have a wife. But it does show that women played an important role in the church, if not in the assembly.
What can/can not a woman do in the church? A woman can not have authority over a man. A woman can not teach in the assembly of the church, except perhaps that she can sing songs (no scripture to support that, though). She can pray in her own home, but not in the public assembly (unless there are no men present). She can raise children and teach them the word of God. She can perform good works, such as benevolence, either on her own or while subject to a man.
Women are the backbone and the working hands of most congregations I know. They are important to the church. That is not diminished by the prohibition against active participation in the assembly of the church.