I would never dream of denying the infallibility of scripture, don't get me wrong. I think that the words of 1 Corinthians are perfect and complete, but our understanding of them may not be. We must seek the right interpretation of these words, IN CONTEXT. That's linguistic context, theological context, and cultural context.
The point of the historical background was to show that we have every reason to believe that Paul himself practiced covering his head when he prayed, even as he wrote these words. What's more, it's equally probable that Christ Himself likewise covered His head with some kind of skull cap, or, even more likely, a tallit. See Numbers 15:38. The garment with the tassels on the corners is the tallit; this is not a matter of oral tradition, this is a "mitzvah," a non-negotiable holy command. We know that Christ wore a tallit because of the occasion where a woman touch it in order to be healed as He walked through the crowd, and He felt the power leave Him. (Luke 8:44, the word translated as "fringe" or "edge" is actually "tassel.") If this was a piece of the old Law that was done away with, I think it would have been given more and direct attention by the apostles, as with other Jewish customs (unclean meat, circumcision, etc.). It is permissible to still be Jewish and practice Jewish customs and still worship the Jewish Messiah, who was Himself Jewish and practiced Jewish customs. Christianity is first and foremost a Jewish religion, a completion of Jewish prophecy, into which Gentiles are welcome to come as well. (I don't mean to sound militant, I'm just trying to put the entire Gospel into context.)
That the man is the "head" (the boss, the supervisor) of the woman, and that Christ is similarly the head of the man, is unambiguous. I am no feminist. Only men should pray in church (unless that assembly is of all women), and only men should teach (with the same exception). Women must submit to the leadership of men, as men submit to Christ. I'm not trying to explain away this passage for political expediency. But I do think it would be theologically inconsistent of Paul to place such a legalistic and superficial command on the body of Christ as a particular article of clothing. Look at the whole issue of the meat sacrificed to idols to see how he transcends petty legalism. Everywhere Paul is concerned with what's in the heart of the believer: a heart full rebellion or a heart of submission, one of works or one of faith. To consecrate or forbid a single article of clothing like a hat sounds to me to be exceedingly Pharisaical and in stark contrast with the rest of Paul's teachings. There must be a deeper, more heart-centered explanation. The context clearly shows that he is speaking of outward showing of respect, submission, authority, and conformity. And since the way in which respect was shown was different even between the divinely inspired author and his audience, we can conclude that it is permissible if the showing of respect is in a slightly different form to our culture.
On what basis do you find it to be "quite clear that this practice of men not covering their heads and women covering their heads in the assembly was the universal practice in all the churches." Is it mentioned in any other letter except this one to the church in Corinth?
