Why doesn't Paul permit woman to teach or exercise authority over man? It's not because of the culture, it's because of the created order and because of the fall:

"I do not permit a woman to teach or exercise authority over a man; she must remain silent. FOR ADAM WAS FORMED FIRST, THEN EVE. AND ADAM WAS NOT THE ONE DECEIVED; IT WAS THE WOMAN WHO WAS DECEIVED AND BECAME A SINNER." (1 Timothy 2:12-14)

God is not the author of confusion. In other words, His Word is simple and easy to understand if you're coming to it honestly, expecting it to change YOU, instead of trying to change IT. There are two kinds of people in the world: Those who seek God's will and choose to obey Him out of Love, and those who seek their own will, and twist the truth of God's Word to suit their own ends.
The Church is it's own culture, God's culture, living as aliens and strangers within whatever culture we find ourselves. We are to be IN the world, not OF it.
Just because the world, which is under the power of Satan, permits certain things and condemns others, does not mean the body of Christ follows suit.
TRUTH DOES NOT CHANGE.
We keep our eyes fixed on our Lord and Savior Jesus Christ, who is the same yesterday today and forever!
Praise God!!