Part 1 is available here.
TWH: Yes. Jesus believed he was God. Let's back up, though, and briefly explain why this question is even asked. The argument is two-fold. First, Jesus never explicitly said, "I am God." Second, there are verses in the New Testament where Jesus' deity is proclaimed. Antonio mentioned seven verses: John 1:1, 18; 20:28; Rom. 9:5; Tit. 2:13; Heb. 1:8; 2 Pet. 1:1. None of these affirmations of Jesus' deity is made by Jesus himself. They are made by disciples of Jesus–John, Paul, and Peter. Since Jesus himself never claimed to be God, we can question whether or not he even believed what his followers attributed to him years after his crucifixion–or so the argument goes.
When scholars say that Jesus never claimed to be God, they are discounting a number of evidences, such as the "Son of Man" references (e.g., Matt. 9:6; 16:28). The very fact that Jesus declares that he has authority to forgive sins is a de facto reference to his deity. And references to the kingdom implies a king who will reign forever (2 Sam. 7:12-13). Jesus presents himself as the Messiah throughout his ministry. He claims God as his Father in a unique sense (e.g., Matt. 7:21; Luke 2:49). And one of the strongest evidences–the connection of Jesus to the glory (e.g., Matthew 16–17).
The argument that Jesus grew into being God is not convincing if you ask me. The early church was wrestling with how to make sense out of this "three persons" and "one God" revelation. Now I don't think this was entirely new to the Jewish religion. You can see what I've written regarding this issue here and here, for example. But they were having to wrestle with it. The disciples, not to mention Jesus' enemies, all understood who Jesus was saying he was when he taught (John 5:18).
Let me just toss out a few more concluding thoughts. The absence of an explicit declaration (i.e., "I am God") from Jesus' own lips does not mean that Jesus did not believe that he was God. There's a huge leap you have to make to get from A to Z with such a deduction. The truth is that even if Matthew had included such a statement in his Gospel, for example, scholars would debate whether or not it was historical and, I bet, most of them would discount it as unhistorical. The word "disciple" does not occur anywhere outside of the Gospels. Shall we say that the apostle Paul never repeated the Great Commission or talked to people about being disciples? I don't think so. The absence of a declaration in a corpus that is limited (sufficient, but limited) does not mean such a declaration never occurred. We just have no record of it. Sure, if Jesus had explicitly said "I am God" the disciples were most certain to have included it. I think Antonio's spot on there. But I don't think the authors of the New Testament thought they were saying anything different from their declarations of Jesus' identity to their presentations of Jesus' life and teachings. The two were saying the exact same thing. For example, what John says explicitly in John 1:1 and 1:18 is consistent with all of what Jesus had said and done in his life. He healed the sick, he forgave sins, etc. He said he was the Son of Man. He told people to follow him (which was to follow God). The same would be true for the rest of the New Testament authors.