Men and Women are different, both physically and mentally. And those differences existed from the creation of distinct genders for humanity in Genesis 2. But the dividing of men and women into different roles in society, especially the notion that men are supposed to be in charge, is purely the product of The Fall in Genesis 3.
In Genesis 2 the Woman is created to be Adam's "Helper" or "Helpmeet". Adam may have been created first but they were still meant to be equals. It is not until Genesis 3:16 that God tells her "I will greatly multiply thy sorrow and thy conception; in sorrow thou shalt
bring forth children; and thy desire shall be to thy husband, and he shall rule
On this side of the cross The Church should be working to restore things to how God intended, though we do so knowing that won't be fully successful until Jesus returns and sets up his Kingdom.
But still we should be working towards that as much as possible. That's why good Christians oppose Slavery in-spite of the Torah's tolerance of it. And why I've argued elsewhere we should oppose Capital Punishment, even though it was an important part of The Law.
The Law of Moses does codify many traditional gender role related laws (but not in the same way most other nations did). But New Testament believers are not under The Law. And even in the Old Testament we had people like Deborah.
Ezekiel's description of the future Messianic Temple has no separate Court of Women just as it no longer has a separate court for Gentiles. Likewise Galatians 3:28 says "There is neither Jew nor Greek, there is neither bond nor free, there is neither
male nor female: for ye are all one in Christ Jesus."
I have another post arguing against those who say women cannot be Pastors/Preachers.
The verses some people dwell on about cross dressing are directed at specific pagan ritual practices involving cross dressing. They were not meant to be taken as legalistically as they are today by many pastors who want to shame women for simply wearing pants rather then skirts/dresses. I notice that particular phraseisim isn't as popular in colder states, like Wisconsin where I live.
The extent to which traditional gender roles are still endorsed in The New Testament (and I believe even NT instructions we shouldn't be legalistic about), are entirely limited to to the relationship between Husband and Wife. Those verses exist purely for the typological application to the relationship between Christ and The Church. Point is, as long as your unmarried those verses have no relevance to you and your interactions with people of the opposite gender, weather your male or female. And if you are married they're relevant only to your relationship with your spouse. Do not go around judging other couples or individuals.
The New Testament certainly doesn't forbid women to be unmarried. Nor does it forbid women, married or not, from having careers or doing God's work outside the household.