I believe the Bible is clear enough on the first question. In Genesis 3, the serpent is the one speaking to Eve and deceiving her. Later, in Revelation 12:9, Satan is called “that old serpent, called the Devil, and Satan, which deceiveth the whole world.” Revelation 20:2 identifies him the same way. So we know Satan was behind the temptation in the garden. Whether he possessed the serpent or used it in some other way, Scripture doesn't tell us exactly how that happened, and I don't believe we need to go beyond what God has told us.
As for why God didn't kill Adam and Eve immediately, I think we need to look at everything that happened in Genesis 3. God did judge them for their disobedience. Sin entered the world, death followed, the ground was cursed, and they were driven out of the garden. God had already warned Adam that eating from that tree would bring death, and that is exactly what happened. They didn't physically die that day, but they would eventually return to the dust.
But something else happened in that same chapter that I believe is very important. In Genesis 3:15, God spoke of the seed of the woman who would bruise the serpent's head. That was the first promise pointing to the coming Redeemer, Jesus Christ. God also made coats of skins and clothed Adam and Eve (Genesis 3:21). Even after they had disobeyed Him, God still showed them mercy.
Romans 6:23 tells us, “For the wages of sin is death; but the gift of God is eternal life through Jesus Christ our Lord.”
So God didn't overlook their sin or excuse what they had done. The judgment was real, and death came just as He said it would. But God also had a purpose to redeem mankind through Jesus Christ. That is what we see unfolding throughout the rest of Scripture.
And I think 1 John 3:8 brings us right back to the serpent in the garden: “For this purpose the Son of God was manifested, that he might destroy the works of the devil.”
