Psalm 51

In Sunday School a couple of weeks ago, we read Psalm 51.  The background is an assumption (I'm not sure it is accepted by scholars) that the Psalm was written by King David after Nathan came to him to tell him he had committed adultery (more on that later) with Bathsheba:

Here are a couple of questions for you regarding the Psalm.

1. "Against you, you alone, have I sinned, and one what is evil in your sight...."  (verse 4a)  My study Bible says, "the harm done to others is by no means overlooked.  All sin, however, affects one's relation to God."  I understand that all sins affects one's relationship with God, but I'm not entirely certain that I see that the harm done to others is not overlooked.  Nathan certainly tried to make it clear, but this verse doesn't reassure me that David "got" it.  What do you think?
2. "Create in me a clean heart, O God, and put a new and right spirit within me." (verse 10).  The Sunday school teacher said this mean that David's heart - and sinners' hearts - are dirty, and they need to be made clean.  Is that what it means?  I wonder if the sinner's heart is broken instead of dirty.  Go on to verse 17b: "A broken and contrite heart, O God, you will not despise.  Doesn't our heart break, sometimes, when we realize how we have sinned and hurt others?  Maybe this is evidence of David's repentance. I think it could be our song of contrition. 

And I'm probably being too picking.  Nit Nit Nit.