What is the main problem in Unwind by Neal Shusterman, and how is Connor affected by it?

I would say that the main conflict that is affecting Connor, Risa, Roland, and other Unwinds is their struggle for survival.  

The book opens by telling readers that the "Bill of Life" allows parents to retroactively abort their child during the teenage years.  The child is "unwound."  This means that every single body part, organ, tissue, etc. is harvested and used for spare parts.  This allows the child to "live" forever in a divided state.  

Connor secretly discovers that his parents are having him unwound.  This is the story's inciting incident.  Connor doesn't want to be unwound, so that is the central conflict surrounding Connor.  He wants to avoid going to a harvest camp because he doesn't want to be unwound.  His desire to live causes him to take all kinds of drastic actions.  The first action is his running away from home and going AWOL.  Along the way he becomes friends with various other Unwinds that are also looking to avoid being unwound.  Connor eventually makes his way to the Graveyard, which is a safe haven of sorts for runaway Unwinds.  As the novel comes to a close, Connor has been given a new identity, which shows him as being too old to be unwound.  He then uses his newfound freedom to become the new head of the Graveyard and help other Unwinds find safety.