Original Sin
Are we born with this? If so why?
It's part of the human condition.
If yes, what did Jesus die for? Did his sacrifice not address this Sin?
Yes. Jesus' death dealt with sin and death. The details of how and in what way vary depending on your particular beliefs.
Here are some of the dominant beliefs:
1. The Calvinist position typically says something along the lines of: Jesus' sacrifice was sufficient to cover all sins, but is only effective for those who are predestined for salvation.
2. The Arminian/Wesleyan view says something along the lines of: Jesus' death makes possible the salvation of all by instilling in every person "prevenient grace"; a grace that, unless resisted, will draw them towards God and lead them to salvation.
3. Universal Reconciliation teaches that Christ's death covers all sin for everyone and all will be saved.
Personally, I tend to focus on views that go beyond atonement and make sense of the totality of everything that was accomplished on the cross. Jesus died to deal with the problem of sin and evil, but not just so that some people might go to heaven after they die. He dealt with the problem of sin and evil in order to inaugurate the Kingdom of God and allow us to reclaim our genuine vocation as a royal priesthood.
If a baby is not baptized and dies prior to, do they die with Original Sin?
This has actually been a very hotly debated topic throughout the ages.
Most today would agree that dead infants may be (but aren't necessarily) saved. Most Calvinists, for example, would propose that those infants who were predestined for salvation are saved and those who weren't aren't, just like with adults. This would include all infants born into a Christian home and may include others who weren't. Some Calvinists go even further than that and say that all infants are predestined for salvation. On the Arminian camp, you tend to find the view that we are born with total depravity, but not condemnatory guilt (Jesus took care of that part on the cross), condemnatory guilt is the result of willful sinning which can only occur after the age of accountability.
Of course for those who believe in universal reconciliation, this isn't even an issue; everyone is saved infant or not.
There are good questions, and the fact they have been debated for centuries attests to this fact. But, I think if we find ourselves focusing too much on this, we may be missing the point. The focus of the gospel isn't to be saved so you can go to heaven when you die. The point of salvation is to empower you to enter into the kingdom of God that currently exists on Earth right now so you can take part in the work of reconciliation God is actually doing. If we find ourselves focusing on the details of what it takes to get to heaven, instead of focusing on how we can be productive members of God's kingdom right now...we missed the point of salvation.