But what interests me more than the head coverings is why these women had to wear them. The context of chapter 11 indicates that women were supposed to wear head coverings when praying or prophesying. This praying/prophesying must've been in the public assembly, else why would they need to wear head coverings? So in my mind the two go together. If women must wear head coverings, shouldn't they be allowed to pray or prophesy in church, as these Corinthian women did?

There is a lot of information on the Mad Women of Corinth: Catherine Kroeger--a feminists--knows some good Greek literature.  Whatever they were doing in 11:5 was not in the assembly: Paul begins to discuss "when you come together" or when you synagogue in 11:17.

The uncovered prophesying was singing, clapping, playing instruments and trying to arouse the customers at the pagan temples. And women in Corinth could be "just out of paganism" where singing-playing women were the public "worship leaders."

While Paul regulates the spiritual gifts it is clear that he uses irony and no one in Corinth was so gifted.  Therefore, prophesying was forthtelling the Word of God.  People might pray in a tongue which was a minor dialect common in Corinth when everyone present could understand Koine.  If one wanted to speak in Latin HE had to pray or speak so that HE could interpret or translate OR he must have an interpreter.

Because it would ONLY be women likely to gibber or sing and play (the meaning of prophecy) if the women were regulated the possibility for an outbreak of what Paul called WRATH or ORGE could not happen among the mature males.   Therefore, here is what happens DURING ASSEMBLY

1Cor. 14:34 Let your women keep silence [includes sedentary] in the churches: for it is not permitted unto them to speak; but they are commanded to be under obedience, as also saith the law.

Paul assuredly knew about Miriam (bitter waters, rebellious) who is marked as a "prophetess" of Hathor connected to Apis the golden calf as the musical idolatry of the Egyptian trinity for which cause God turned them over to worship the heavenly stars. The word for "prophetess" is defined as "an Egyptian Sorceress." And interestingly most of the singing and instrument names point to enchantment: John called them sorcerers in Revelation 18.  This is just NOT something most pre postmodern "males" would fall into other than the leader of the charismatic band who ceased being charismatic when they fell out of love with the "one Warlock out of 10,000 witches."

There are some things left as signs or MARKS: "because all religionism is promoted by disenfranchised or marginalized males and their chief supporters, the women, and males never fall into it" it has been said especially of the wild frontier revivalism, if you have an invasion of the "musical body snatchers" you can be guaranteed that there is a TOP ELDERESS doing the leading. Sorry, history knows no exceptuion.

There is another MARK of the Warlock: "if he does not speak according to the Word there is no truth in him." Or of the elders as the ordained preacher-teachers "he must teach that which has been taught." Peter left that so we can MARK the false teachers just as false prophets were marked.

The question was asked
Should women cover their head in church?

The answer is

Yes if they are ugly please cover the head

Big Mike Lewis wrote:
Finally, someone who really knows Greek well enough to take on piney's weird posts.

Big Mike,
His abuse of the Greek is easy enough to handle. The bigger challenge is making sense of his English. 
Melchizedek