Yes, Paul was a man of his times and, I suppose, he cannot help but reflect his culture. However I reiterate, cultural standards were not his emphasis. A lot of what he did was troubleshoot problems that cropped up in the local churches.

Without getting into theology 101 and treating every point (which I confess, sounds like fun to me) let me just say for now that there is a branch of study known as hermeneutics and it has to do with the internal logic and consistency of the work at hand. Paul was a brilliant and educated man in his day. That his own understanding reflected the culture(s) in which he was immersed (pun intentded) is wholly understandable and at the same time demonstrates the authenticity of the times in which he wrote.

Paul, via his amanuensis(es) wrote to Jews of the Diaspora who were well versed in the Jewish culture and he drew allusions to there understanding of precedence and order as found in the Torah, the male preceeding the female. This wasn't such an issue with many of the gentiles. He said salvation was to the Jew first then the gentile, the order reflecting that Christ came to the Jews first and then was rejected.

Paul's real mind on the subject? For anyone who could get beyond the cultural differences of their day, read Galatians 3:26-29, especially 28 "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'd call that pretty forward looking and transforming. Those ideas helped reshape the world and uplifted the sstatus and role of women everywhere.