Just some general thoughts, which unfortunately may be worth nothing at all. But, just in case...
Satan the Serpent
The thing about the serpent being satan was brought up several times. Christians believe the serpent was satan because the Bible says so twice in Revelation (12:9 and 20:2), not because of logic deduced from the story. Paul also said the serpent "beguiled Eve through his subtlety," suggesting he would have considered the serpent at least a tool of satan (2 Cor 11:3). Most Christians have never thought about it beyond this. The serpent was satan, and it's as simple as that.
I doubt that Christians made that up. I don't know what the modern Jewish view is of the serpent, and I'm sure Jews don't take the story literally, like Christians do, but I have heard some things about satan in Jewish history that would apply.
A satan was an "accuser" or "adversary." Originally, I understand there could have been many of them. The satan of Job walked in heaven on a regular basis. He was most definitely hostile to God, but he was not some banned creature that had led a failed rebellion in heaven.
I think ancient Jews would have considered the serpent a satan simply because he was an "adversary," an opponent of God. The word satan eventually applied to just one being, and the thoughts about the serpent were never really updated.
The Lying Serpent
Even when I was a young, brand new, on-fire (Bible-toting, Scripture-quoting, devil-chasing, sin-defacing, etc.) Christian in the early 80's, I could never figure out why I was being told the serpent lied. I asked numerous people what they meant by that, and I never got an answer that made sense to me.
It's the same with the commonly stated opinion that satan misquoted Scripture to Jesus during the temptation. I don't see that satan "took the quote out of context" at all. Yeah, he applied it where Jesus didn't think it should be applied, but the quote was accurate enough.
Anyway, I doubt that was worth even two cents, but just in case.