Can women be catechism teachers in the Catholic Church?

Yes, they can, to children. But keep in mind Paul's principle, 'I do not permit a woman to teach or have authority over the man,' which applies not only in the family but also in the church; that is why women cannot be priests, because they cannot have authority in the church, and it carries down to the lower levels of teaching as well. This is why all the other churches are having problems with the role of women: if you do not cut that line sharply, they will say they can preach sermons and teach and do everything a priest does, and before long they are. In the Catholic Church we laid down this law long ago and nobody has transgressed it, thank God, whereas the Lutherans, Anglicans, Calvinists, Methodists, and Baptists transgressed it long ago and have even elected a woman as Archbishop of Canterbury. Women are still, in a sense, being punished for the sin of Eve, which is why Paul in 1 Timothy 2 says a woman will be saved by childbearing, because that is the opposite of what Eve wanted, which was to become a god and not have anyone, God or her husband, tell her what to do.