Well you are pushing for the favor of Islam to the point that it seemed like you were almost saying Muslims always existed along with the Jews. Besides that I understand your point about Ishmael but Muhammad wasn't a Prophet when compared to his great ancestor or any other of the Prophets before him for that matter.
But I'll leave it at that since it clear to me you don't consider looking into what I have to say about them being different. Instead I'll just answer your very first question about the children of God thing.
48 He replied to him, “Who is my mother, and who are my brothers?” 49 Pointing to his disciples, he said, “Here are my mother and my brothers. 50 For whoever does the will of my Father in heaven is my brother and sister and mother.” -Matthew 12 NIV
and
For those who are led by the Spirit of God are the children of God.
-Romans 8:14 NIV
and
26 So in Christ Jesus you are all children of God through faith, 27 for all of you who were baptized into Christ have clothed yourselves with Christ. 28 There is neither Jew nor Gentile, neither slave nor free, nor is there male and female, for you are all one in Christ Jesus. 29 If you belong to Christ, then you are Abraham’s seed, and heirs according to the promise.
-Galatians 3 NIV
This is why I asked who told you that as the Bible is clear who is the children of God are.