Humans are tribalistic. We form hierarchical societies spontaneously, so I don't think anarchy would last long before being replaced with some form of government. If anarchy could be sustained then I think it would fail for the same reason that every other system of government has failed: We are inherently are selfish.

Crime is punished to disincentivize crime. Unfortunately, high-profile criminals often outrank the courts of law. I think that good education and parenting are excellent methods for decreasing crime. Children who understand morality will often carry those morals throughout their life, or even expand upon them   I suspect that a child who is treated with fairness or respect is more likely to treat others with fairness or respect.