Is God a divided personality, one of justice and one of mercy?
God is just, merciful and loving; because he IS love.
Why does God need an innocent God to die before he can forgive?
Mankind sinned against God.
As the One who was sinned against, God had the right to say what people needed to do to atone for their sin. Possibly you might say, "why did that have to involve sacrifice and the shedding of blood?" But it wasn't our call to make.
The Israelites were given God's law but were unable to keep it - they were offering more and more sacrifices to cover all their sins. So eventually God made a NEW Covenant. He sent Jesus to offer himself - to lay down his perfect life - as a sacrifice for our sins.
We cannot DO anything to atone for our sins against God. So we can never deserve to be forgiven and reconciled to him.
The Bible says that the wages of sin is death = spiritual death; eternal separation from God. THAT is what we deserve. But because God is love and merciful, he said, in effect, "I will take the punishment for your sin" - and he sent Jesus to die for us.
It's like a judge in a court passing a guilty sentence, saying "the punishment for this crime is ........I know the defendant won't be able to pay that, but the law must be obeyed - so I'll pay it.
This is why John says, "THIS is love; Jesus Christ laid down his life for sinners".
WE couldn't pay the price for reconciliation to God. So instead of saying "tough luck, you're all going to hell then"; he paid it for us.
Why did Jesus teach a gospel of salvation by faith and forgiveness before he was killed for teaching that gospel?
He wasn't killed for teaching people to believe in God and forgive one another; the Pharisees wouldn't have objected to that.
He was put to death because he claimed to be God. He was telling the truth but the Pharisees etc didn't believe him and had him killed for blasphemy. If you read the Gospels, you'll see they tried to stone him twice for that reason.
If the Jews would have accepted Jesus, the entire purpose of their faith, and accepted the original gospel, what would they be teaching today from Jerusalem?
Some Jews did accept Jesus and believed, and taught, that he was the Messiah.
There are Jews today who believe this, and are coming to believe it. They are Messianic Jews. I guess some of us would say that they have converted and become Christians, but that is not so. Christianity has its roots in Judaism, and Messianic Jews are, I believe, complete Jews. Jews who believe that their Messiah - prophesied in the OT - HAS already come, rather than waiting for him to come in the future. But you'd need to find the forum for Messianic Jews on these forums, to find out more.