Inspired by what I think might be a gap in the answers to this question, I wanted to know from those who study early christianity exactly how the devil became the devil. Why is the Christian scripture satan so different from him of the Jewish tradition?
My guess from what I have learned (in cursory fashion over the years) is that eschatology had much to do with the shifting Satan. Apocalpytic anxiety and fear lends itself to questions of judgment and hells. We see this in Chinese Buddhist and Daoist movements of the same period (1st-5th centuries CE) as christian movements. So, did eschatology help define the new Satan and hell that emerged with Christianity?
Bart Ehrman is a scholar who has written extensively on this subject. Jesus: Apocalyptic Prophet of the New Millennium covers many of the questions that you stated. Its a very engaging book for being a scholarly work and I highly recommend it to anyone.
Because I am not an early christian scholar, I will quote directly from his book after putting these quotes in the proper context.
In Jesus: Apocalyptic Prophet of the New Millennium, Ehrman explains that apocalyptic teachings had really ramped up leading into the first century c.e. Both Jews, and later Christians were obsessed with death and, like you hypothesized, satan and the way his place inside the religion shifted.
"Around the time of the Maccabean revolt, when the oppressive policies of Antiochus Epiphanes became too much for many Jews in Palestine to bear, when they were forbidden on pain of death from keeping the Law of Moses, some of them came up with another solution. In their view, the suffering of God’s people could not be explained as a penalty for their sin. God surely would not punish his people for doingwhat was right, for keeping his laws, for example. Why, then, did the people suffer? There must have been some other supernatural agency, some other superhuman power that was responsible. God was not making his people suffer; his enemy, Satan, was.
According to this new way of thinking, God was still in control of this world in some ultimate sense. But for unknown and mysterious reasons he had temporarily relinquished his control to the forces of evil that opposed him. This state of affairs, however, was not to last forever. Quite soon, God would reassert himself and bring this world back to himself, destroying the forces of evil and establishing his people as rulers over the earth. When this new Kingdom came, God would fulfill his promises to his people. This point of view, as I have said, is commonly called apocalypticism." (pp. 120-121)
"...Jewish apocalypticists maintained that the entire creation had become corrupt because of the presence of sin and the power of Satan." (pp. 122)
Ehrman, Bart D. (1999-07-26). Jesus: Apocalyptic Prophet of the New Millennium. Oxford University Press. Kindle Edition.
Erhman isn't the only scholar to make the claim (with both biblical and historical evidence) that the story of Satan in the world really began to change near the turn of the millennia. John Dominic Crossan and Albert Schweitzer both have mentioned this same idea in some of their works.
Hope I helped