Thanks for two excellent questions. Unfortunately I’m only going to have space here to deal with the first one properly. You could ask your second one again though (by itself) to get an answer.
I think the verse you are referring to is Ezekiel 18:20: “The soul who sins shall die. The son shall not suffer for the iniquity of the father, nor the father suffer for the iniquity of the son. The righteousness of the righteous shall be upon himself, and the wickedness of the wicked shall be upon himself.”
It does seem just to punish only the one who actually does anything wrong, doesn’t it? We Christians, somewhere deep down, want to agree with this sentiment wholeheartedly. But there is a problem - “The soul who sins shall die”. The problem is that I’ve sinned, you’ve sinned, we’ve all sinned. We’ve all done wrong things and according to the bible we all deserve to die (check out Romans 3:23). Setting the question of original sin aside, we will be punished for our own sin first!
But how can we escape? Only because someone else was punished for our sin. If you keep reading that Romans passage then 3:25 says that God offered Jesus as a “propitiation” or “offering of atonement” for our sin. These are big words that mean that Jesus took the punishement that we deserve and died in our place. In your words, Jesus “bore the sin of another man”. In fact, many. Have a look at Is 53:4,6:
“...surely he has born our griefs and carried our sorrows… and the LORD has laid on him the iniquity of us all…”.
In fact, the whole Christian worldview is utterly dependant on the concept of “sin bearing”. You might like to check out Leviticus 16 as well sometime - to see where the line of thought starts.
This is important, but it still hasn’t answered your question. Christians believe that what happened when Adam sinned was not just Adam “doing the wrong thing”, and we get punished for it. That’s not it. Rather, something fundamentally changed about Adam’s nature. He didn’t just sin, he became a sinner. From then on, Adam sinned because he was a sinner - “by nature a child of wrath” (Eph 2:3). Since we are his children, we share in his nature. That is what is means to be human in a post-fall world. You sin because you are a sinner. That doesn’t get you off the hook - sin is still wrong. As that Ezekiel verse said: “The soul who sins shall die.”
I guess the only thing we can say is thanks be to God who loved us so much that he sent his son while we were “dead in our sins”, to bear our sin and take our punishment, so that we could be “made alive together with Christ”. (Eph 2:5)