The important thing here is to remember what Paul knew very well about Timothy –
Timothy the bishop learned the faith from his mother and his grandmother.
If you turn to 2 Timothy 1:5, Paul says straight out:
“Calling to mind that faith which is unfeigned in you, which also dwelt first in your grandmother Lois, and in your mother Eunice; and I am certain that it is in you also.”
In the same letter, Paul passes along greetings from Claudia, and most of his letters pass along greetings from women. So it’s hardly a case of “All you ladies, shut up and stay out of the way.”
Moving along… most Catholic interpretation says that “saved through childbearing” is a discussion of how Mary’s obedience of God is the opposite of Eve’s disobedience of God, and how Mary and the Church both “bear Christ” (because the Church makes new little Christs, ie, new baptized Christians).
(Paul seems to talk around and about Mary without actually mentioning her, always in a respectful sort of way. (Some people suggest it’s because she was still alive at the time, and Paul didn’t want her arrested or harassed; some people say it’s because it was a big mystery.)
So “the woman” Eve, who is also Israel and the members of the Church here on earth, messed up through sin. But “the woman” Mary helped save us through childbearing of Jesus; and the members of the Church (male and female) all have to “continue in faith and love and sanctification, with sobriety.” This goes right back to what Paul already said about everyone in the Church, male and female, and why we should pray for secular earthly leaders: “So that we may lead a quiet and a peaceable life in all piety and chastity.” Women aren’t the only ones supposed to be quiet and chaste. That’s everybody.
(Why, yes, Paul’s letters tend to be very “stream of consciousness.” And if you think he’s bad, try St. Augustine, the king of ADHD and making side comments. )
What’s generally told to us is that a lot of women (and men, but apparently women did it more often) who were prophets would sometimes start prophesying right in the middle of Mass, at the top of their lungs. This was a no-no. There’s also a lot of men mentioned in the letters who went off and started preaching weird stuff, and they get a talking to, because you can’t go around preaching and ordaining yourself when the Church hasn’t appointed you, and because they say weird stuff. Some women apparently also did this, and they get an additional talking-to because women can’t be ordained, much less ordain themselves. (Because Paul’s already talked plenty about people preaching blasphemy and weird stuff.)
Paul does tend to talk more to women about how they dress and how they fix their hair. (Although he does talk about this to men in some of his letters. Greek men really liked long hair and perfume and stuff.) But that’s because the standards of dress for rich, adventurous Roman and Greek women who wanted to explore a new religion were really, really different from how Jewish ladies or poor Roman and Greek women would dress. It was wasteful to have too much luxury in how men or women dressed, and it caused envy and distress among the poorer members if you showed off too much.
It could also give people the wrong idea (and did!) about these sexy dressed men and women, both going to those secret meetings that Christians were always attending. (Yeah, a lot of people thought a nice agape dinner was the Christian way of saying “orgy.” After all, there were all these women hanging out with men! And all these sexy long-haired Greek guys wearing perfume!)
It was a period of big social change and some religious cults did have orgies! So Paul had to lay down the law a bit, not that it really helped once the urban legends about Christians had gotten started. (But he hardly regulates anything at all, if you compare his stuff to the intricate regulations laid down by Jewish rabbis for Jewish women. Hokey smokes!)
Paul had high expectations of women. We know that his teacher, Gamaliel, had a learned daughter who is quoted in the Talmud. Nor was he apparently shy of working with women in the Church. I’ve never felt he was against me, as a Catholic woman.
Finally, if you read one of the pagan sneers against Christianity, you will quickly find out just how appalled they were by all the freedom and niceness and favorable attention given to women. (And kids. And slaves. And barbarians. Boy, those Christians.)
Paul doesn’t have anything against women, or men, or anybody. If anything, he seems a bit uncomfortable with telling people what to do, because they should ideally already know how to act “in the house of God.”
PS. There’s a bit more about women at 1 Tim. 3:11, and you’ll be able to see there that he’s talking about the wives of people getting picked as deacons (or bishops). Obviously it’s what all Christian women were ideally supposed to be anyway; but if you’re going to have a guy working all the time for the Church in a position of responsibility and discretion, his wife has to be trustworthy too. (The deacons handled going around the city and giving out the alms money to the poor, and their wives gave it out to poor women who didn’t have men in the household. Occasionally a deacon’s mom or sister did it, or an old church lady. Depended on the individuals chosen, and whether they had wives or not.)