With regards to the first part of your question, we need to first draw a distinction between the claims Jesus made about himself during his own lifetime and the claims that his followers made in the decades after his death. Scholars are largely in consensus that Jesus never claimed to be divine, much less God himself, during his own ministry; this idea, which out of the four canonical gospels is only found in the latest, John, emerged in some Christian circles towards the end of the 1st century.

So what did Jesus claim during his own lifetime? Based on the traditions about him preserved in the gospel accounts (that pass the scholarly criterion for historicity), Jesus identified himself as the Jewish Messiah, the one whom God had chosen to rule over the new Kingdom of God that he would very soon establish upon the earth. This is especially evident in the fact that all the gospels agree that Jesus was put to death by the Romans for claiming to be “the King of the Jews.” Jesus was not the only person to have made this claim; Simon bar Kokhba, who led the eponymous revolt against the Romans several decades after Jesus’ death, at the very least embraced speculation by his followers that he was the Messiah. However, Jesus’ legacy survived long after any other claimant for a very specific reason.

Why did Jesus’ followers accept his claims to be the Messiah? Well, it wasn’t because of how his ministry ended; no Jew prior to the death of Jesus believed that the Messiah was going to suffer and be crucified. Jesus disciples followed him during his ministry because they believed, so scholars conclude, that he would be the exact sort of Messiah they were expecting; a king who would rule over the imminent Kingdom of God on earth. Jesus even promised them positions of power in this future kingdom, declaring that when the Son of Man arrived in judgement, his Twelve Apostles would “sit on twelve thrones, judging the twelve tribes of Israel” (see Matthew 19:28). His death threatened to shatter the movement he had created, as it seemed to dash all his followers belief in his claim to be the Messiah. That changed for one specific reason, which I mentioned earlier; his followers came to believe that Jesus had been raised from the dead.

According to Paul, the earliest source we have on this subject, after Jesus death his disciples experienced visions, in which they saw Jesus exalted to heaven. Paul recounts that “he appeared to Cephas, and then to the Twelve. After that, he appeared to more than five hundred of the brothers and sisters at the same time…then he appeared to James, then to all the apostles, and last of all he appeared to me” (see 1 Corinthians 15). This fact, ultimately, is what made Jesus followers continue to accept his claims to Messiahship (if not divinity) even after his death. It was the simple fact (so they believed) that he had been raised from the dead and exalted to heaven, and God certainly wouldn’t have done that for a false messiah! We should take Paul’s own “faith trajectory” as instructive here. Paul at first found the Christian message so objectionable that by his own admission, he became a persecutor of the early church. What eventually changed his mind was his own visionary experience of Jesus; it was this visionary experience, and others like it, that made Jesus followers accept his role as Messiah even after his death.

So how did that Messiahship evolve into outright divinity? It’s not an easy or straightforward process to work out, but Jesus already seems to have had a semi-divine status by the time of Paul, who speaks of Jesus as pre-existent, having been with God in heaven even before his birth (see Philippians 2:6). It’s important to note that divinity for ancient peoples was a diverse spectrum, with some beings being more divine or less divine than others; angels, for example, were more divine than humans, but not as divine as God himself. Paul seems to have conceived of Jesus in a similar fashion. From here onwards, claims about Jesus became more and more exalted over time, until we get to the Gospel of John, where Jesus is stated to be one with God the Father. This process did not occur in a linear fashion, and Christians held diverse opinions about the nature of Jesus for centuries, some teaching that he was purely human (adoptionism), that he purely divine (docetism), or some combination of the two. Ultimately, however, the recognition that Jesus was truly “God” won out among the majority of Christians, and became the teaching team expressed in the Nicaea creed, defining the positions of the Catholic and Orthodox churches (among others) down to the present day.