Well, I'll do my best. Although Jewish, I have certainly tried to study Christianity, and I think I probably know more about it than even the average Christian. But I am by no means a scholar, and certainly have no ability to read Koine Greek.
I think this is above my paygrade, as it requires knowledge of Greek.
I'm honestly not even sure what Peter meant by Messiah. He was Jewish, and in Judaism the messiah is a human, earthly king that rules Israel during the messianic age. But Christians developed a completely different idea of Messiah to mean God incarnating and dying for their sins. So I can't be sure what Peter actually thought.
Its far too late and I'm far too tired to go into detail, but generally speaking, I am of the opinion that the earliest Christians, the Jewish believers in Jerusalem under the leadership of James, also known as Nazarenes, did NOT believe Jesus was God. It was when Paul took the gospel to the Gentiles that people who had a Hellenistic world view made Jesus into a deity with Paul's encouragement.
Oh, I'm surprise to hear you propose that. I don't see Jewish influences in the Nicene Creed at all. The Tanakh teaches four times that God is not a man. The idea that a man can be god is completely pagan, and it entered the church via Hellenized non-Jewish converts.
I don't see why you can't address "Jewish thinking" at the time of Jesus if you wish. We know there were various groups in Second Temple Judaism (is that the label you were looking for?): Pharisees (bet Hillel and bet Shammai), Sadducees (mainly priests), Essenes, and Zealots. When the temple was destroyed, the Pharisees were the only group that survived, making their views the only views in Judaism.
A pleasure speaking with you.