What was the Roman world like at the time of Jesus' birth?
In the century before Jesus' birth, the Romans were just ripping each other apart. Until a few decades before Jesus was born, the Roman system had a marvelous administrative way of avoiding tyranny by not having a king, but by putting in two consuls, who were like kings for a year. Each of them kept an eye on the other. That system had worked fine for the Romans for centuries.
But as soon as they got an empire in the century before Jesus' birth, the two consuls started to grab more and more power, and they weren't going to share. They were in transition from a republic, which had worked for 600 years, to an empire. They took the mightiest military machine of its age, split it in two, and sent it against one another, so Italy was being ripped apart, Greece was being ripped apart, the whole Mediterranean world was being destroyed. When finally Caesar Augustus won out--he was the last one standing--there was a huge collective sigh of relief from all over the Mediterranean.
This was the 2nd of September, 31 B.C.E., 31 years before Jesus' birth. At that point, the elite said, "It's over, it's over, peace." The economy started booming again, the sea lanes were free of pirates, the roads were relatively free for commerce. So Caesar Augustus became a god incarnate.
Today, if you talked to most people and said there was a human being in the first century who was called Divine Son of God, God from God, Lord and Redeemer, and Liberator and Savior of the world, 99.9% of people would say it's Jesus we are talking about. But Caesar Augustus was called all of those titles before Jesus was ever born. Those were his titles.
Now, there were many eternal gods, those who had been there forever, like Jupiter--in Latin he would be called a "deus." And there were divinized humans gods like Caesar--in Latin he would be called a "divus."' If you were a human being who had done something very very good for the world, you were elevated to the status of a god. People at that time believed it could happen because they also believed gods and goddesses came down from Heaven and married or had intercourse with humans and produced a child who was a semi-divine being.
But in Greek there is no distinction between `deus' and `divus', so if you are speaking Greek, when you say "son of God" it seems to mean the same thing, whether you're referring to a divinized human or an eternal god. There is no other word like a "Son of a Divine One" in Greek. So when this translates into Greek for Jesus, and for Augustus, they are both, in Greek, "Son of God."
Fully divine. The Romans didn't waste much time saying, "How can Caesar be fully human and fully divine?" They did it primarily with images. On every coin you have inscriptions of Caesar as divine. In the ancient world, being divine was a job description, meaning somebody who does something very important for the human race. So when that claim was made of Jesus, they were not claiming that he wasn't human--they were quite aware in that world that he could be a human being--but they claimed it based on his having done something extraordinary for the human race.
Did Jesus believe himself to be this title that he was later given by the Gospel writers and by Paul?
Jesus was claiming to know the will of God and to know what the Kingdom of God was supposed to be. If I were talking to Jesus, I would say, "How do you know that? How are you so certain that Caesar isn't right? He's brought peace to the world and everything else, how do you know your system is right?" I think he would say, "Well, it's in the covenant with God." If I asked if he was the Messiah, he would ask, "What do you mean by Messiah?" If the Messiah is the leader who is going to win a battle against the Romans, then no, Jesus wasn't the Messiah. But if the Messiah is the one who reveals the word of God, Jesus' answer might be, "Yeah, because I've done that."