Sin is essentially a rejection of (or deviation from) God (or His ways). I tend to think of the event of Adam's disobedience as the "original sin", which resulted in mankind being separated from God... as opposed to mankind possessing a particular kind of ("original") sin due to Adam's mistake - if that makes sense.
In other words: We are all born separate from God in the sense that He is not here with us in communion and fellowship with us, due to the Fall. It is in this sense that we "inherit" separation from God, which could be phrased as "inheriting sin", or "inheriting original sin" (although the latter is slightly confusing.)
In a state of separation from God it is impossible to do good (God's definition, not man's), and so every action is essentially sinful, being independent, selfish, etc.
If man dies in this state of separation, he dies eternally separated from God, which we call Hell. (This of course has to be balanced with an understanding of God's justice, but I'll let that lie for now.)
So to summarize the first part, our inheritance from Adam is separation from God - a state from which every action of ours condemns us to judgment.
The question, then, is "how is that just?" Essentially we are all headed for the judgment of God because of what Adam did, right? Actually, this would be true if that were the end of the story; if God judged solely based on "original sin". But that is not the entire story.
Adam sold his family (us) into slavery to sin, and now we all sin from childhood - disobeying parents, stealing, lying, hurting each other, rebelling against God, etc. Although we will not be judged for what Adam did, we will be judged for what we do, even if it is the result of our fallen state, which is the result of what Adam did! (Hang in there...!)
Of course, God couldn't leave us in this pitiful state - He is too loving and too just. As a result, God provided an opportunity to return to Him through Christ. Now it is possible for a man with a wicked father to turn to Christ and be saved from judgment, despite his father's actions.
Now we have to touch on a much more complicated topic: election. See here for a much more complete explanation, but essentially, God knows in advance who would embrace Him and who would not, and "chooses" His people based on this foreknowledge. He then intervenes and redeems those whom He has chosen.
When all is said and done, here is the picture painted by Scripture:
The "righteous" will be justified by their own deeds, which are of course accomplished in partnership with God. These folks only become "righteous" by the grace of God, according to His foreknowledge of their willingness to be so changed.
The "wicked" will be condemned by their own deeds, which are by definition deeds of independence from God. The wicked would be wicked whether they were placed in the garden prior to sin or born "fallen." In God's justice and impartiality these folks are not wicked solely because God didn't intervene; they would have rejected Him regardless of what opportunities He gave them.
So in the end a man will be justified or condemned by his own deeds alone (which are ultimately the result of his acceptance or rejection of God). That is where the passage you cited comes in. Ezekiel 18:20 is a passage about God's justice. He is correcting the impression that children suffer judgment from God because of the mistakes of their fathers, but God is clarifying that a person is only judged by their own actions.
The point of the passage is that God is just (which He is, as we have just seen). It's difficult to understand without seeing the big picture, because at first glance it looks like Adam sold us out and we're all "up a creek" so to speak. But God's justice was not thwarted by Adam's decision - ultimately He works it all out properly according to His infinite wisdom.