Good post, @DamianTV!
There is an inherent contradiction that we can't escape... Let me try to put it this way:
Most, if not all pre-colonization native american tribes had way less crime, less murders, than in our society, and they had no government... Now you might think; "Really? No government? They had chiefs, councils and waged war..." The truth is that the chief had in his position no real power, no force. He was a wise man that they listened to, he held no coercive power, nor had he the warriors under his command. He collected no taxes. If the young men of the tribe wanted to go to war, he could not stop them, all he could do was to urge caution. In a similar manner, he could not initiate war, he had no resources - unless he could persuade the majority opinion of the tribe in his favor. (as any person could do)
But yet, the indian tribes had no technical developments over thousands of years - because they had no private property - everything was shared. If the hunt was good, everybody ate well. If the hunt was scarce, everybody went to bed hungry. No private property, except clothes could exist, because there was nothing significant of value that could be stored for significant amounts of time. Now if say they had started growing large tracts of lands with agriculture, (many crops can be dried and stored for years) the most efficient way to administer that would be to have private property, in a sort of confederacy of independent farmers. But crops often fail as well, leading to hunger and starvation in primitive societies. Would not the ones that had starving children band together and raid/kill the few farmers who's crops were successful that year? - Yes, since they are desperate. Thus it seems that crime will increase as one moves from the most primitive methods of production to an more advanced one, leading to the destruction of private property, and eventually one will go back to primitive communism, since the incentive to work very hard to produce large surpluses will not be there if it is all taken away in bad times. That is, unless you install a government, some kind of court that has a monopoly on power that uses it to protect private property. Crime will still exist, but not be significant enough to make society fall apart. Thus technology and society can advance to produce larger and larger surpluses.
I will conclude that government is definitely a prerequisite for larger society and more advanced forms of production than "subsistence-hunting-communism". At the same time, governments never prevent crime in themselves, they can only deter some people from committing them by promising punishment. (obviously)
If we had a society that produces even larger surpluses than now, so that the poorest people then, will be people we consider well off today - there would be less crime, but it would still exist - look at some of the largest corporations breaking all kinds of laws because some greedy people want it all. (So even if all needed social help for disabled people are provided through familiy and charity, so that there would be no need for government... except for that it's primary function would still be needed - the nessesary evil that is monopoly on force to protect private property.)
//I think parts of what I wrote may have been a bit "rample on", but I hope you understand the gist of what I'm saying.