1 Corinthians 11:3 has been used, especially by Muslims, to “prove” that Jesus is not God according to 1 Corinthians, but their argument is weak at best and it ignores clear and unambiguous statements by Jesus and others that Jesus is God. Here is why this passage does not say that Jesus is not God. “But I want you to realize that the head of every man is Christ, and the head of woman is man and the head of Christ is God.” This passage is not about who is greater than another, but who submits to another. If this scripture proves that Jesus is not God, then it also proves that women are not human beings. The relationship between man and Christ is analogous to that between woman and man and between Christ and God. So, how are all three of these related? They are related by positional submission. Women and men are equal in the eyes of God (Gal 3:28-29). But woman submits to man, presumably in marriage (Ephesians 5:22). Similarly, Jesus is equal to the Father, but Jesus submits to the father (John 14:9-10). Given the analogy between the husband/wife relationship and the Father/Son relationship in this passage, it cannot imply that Jesus is not God. Of course, this passage also does not prove that the Son is equal to the Father, as it also includes the relationship between man and Jesus, but this is a bogus proof-text for proving that Jesus is not God. The same can be said for 1 Corinthians 15:28, with a slight difference. In 1 Cor 11:3, given the analogy it uses between husband/wife and Father/Son, it is definitely wrong to use this passage to support the conclusion that Jesus is not God. In the case of 1 Cor 15:28, it does not mention the analogy, so it is not absurd to try to use 1 Cor 15:28 to support that Jesus is not God. However, given its similarity to 1 Cor 11:3, and given that Paul made both statements, we can conclude that it is also very dubious to use 1 Cor 15:28 to prove Jesus is not God.
Given your second question, I can now tell that you are responding to Muslim criticism of Christianity, as only Muslims use the first two arguments, as far as I know! In Islam and in the Qu’ran, prophets and great men and women of God are presented as nearly sinless. Muhammad is treated as if he were nearly sinless (which he definitely was not if you study his biography!). To the Muslim, a great prophet cannot take part in gross sins. There are two problems with this Muslim criticism of the Old Testament. First of all, who says that God cannot use a person who has sinned in the past to do great things? This is a presupposition that Muslims are taught, but it is simply that–a presupposition. The Bible DEFINITELY does not agree with this presupposition, and I can think of no reason that God cannot use sinners to do his will. Second, the facts we have about the great people, both men and women, in both the Old and the New Testament proves that this assumption that God cannot use a great sinner is simply false. Abraham doubted God and lied about his wife. Noah got drunk. David committed adultery and murder. Paul sponsored the murder of Christians. Peter denied God. Jacob lied and used that deceit to gain his father’s blessing. As for Lot, he was not a prophet of God. He is never called a prophet in the Bible and he never does any of the things a prophet does in the Bible. Like David, he committed detestable acts. By the way, just so you know, I have committed detestable acts as well. I am guessing that you have as well. Who am I to judge Lot? If God chooses to use Lot, who am I to question God? This criticism of Muslims is an unfounded and unreasonable attack on the Bible.
This third question, unlike the first two (in my opinion) is a good and reasonable one. Scholars are not unanimous that Jephthah actually killed his daughter. I believe the minority believe that his “sacrificing” his daughter meant that he actually killed her. What the Bible actually said is that she committed to never get married. A better conclusion is that she became something like a nun–a woman committed to God and who does not get married. God definitely hates human sacrifice. If this is the proper interpretation of what happened in Genesis 11, then the question is solved. Personally, I believe that this is the case.
Having said that, some–and even some faithful Christians–believe that Jephthah actually sacrificed his daughter by having her killed. I do not agree with this, but what if it were true? Would that be a clear contradiction with what we know about God? I am not sure. Obviously, God hates human sacrifice. Yet, God gives us free will and he allows human beings to do terrible things such as genocide and, yes, human sacrifice. God allows things that he hates very much because he gives us free will. I believe that this is not the correct understanding of Judges 11 (that Jephthah killed his daughter), but I certainly can understand why people would be really troubled if God allowed such a terrible thing to be done by Jephthah. I can also see how this could be difficult for people to find coherent with the other things we know about God. If Jephthah killed his daughter, given that his other acts are presented as a good example, that would be confusing at best. Yet, the Bible is very “real.” The Bible presents its great men as truly flawed people. So, although I disagree that Jephthah had his daughter killed, I am not sure that there is a clear contradiction with what we know about God that he might allow such a terrible thing to be done by Jephthah.