It's easier to take those questions in reverse order.

What isn't sin is the nature, person, and work of the Trinitarian God as outlined in Scripture. This is what we may refer to as "holiness" or "righteousness" or sometimes even "glory". Notice that, just as it is with us, the actions of God flow from His character. Righteousness is NOT the summation of all of God's actions (what He does), but rather the sum of His nature and traits (who He is).

Because holiness is the sum of character, not the sum of actions, sin is not something you do, but rather something you are. Contrary to "Bible Belt" religion, you perform rebellious actions because you are sinful. Religion (in the bad sense of the word) flips this around, preaching the false doctrine that we are only sinful as we sin. But sin is a much deeper issue than actions or deeds. Sin is a heart issue. It deals with what Jonathan Edwards (and others) refer to as the affections, or the passions.

Sin is not something you do, but something you are.

This manifests in different ways, primarily through rebellion (I'm God over my life here and now) or through religion (I follow God here and now so I can be God later).

But thankfully, there is an answer. Jesus didn't just do all the right things...He became all the wrong things. Jesus did not just go through a transfer of actions, but of characteristics.

Dwell on this verse, and how it uses the verb "be/become":
For our sake He made Him to be sin who knew no sin, so that in Him we might become the righteousness of God. (2 Corinthians 5:21)

Hope this helps. I'll be glad to answer any follow-up questions or point out further resources.