That's an interesting idea Sic. If there was such a massive abundance of resources, would anyone bother to compete? I would argue that yes, people would still naturally compete, because there are things other than food that people value. For example, people would compete for lovers because presumably attractive mates can't be an abundance... you can always do better. Everything can't be in abundance, even if our basic necessities are, and people will compete for those things if it betters their life or increases their reproductive success.
Abundance may exist at some times, but it's always ephemeral because abundance encourages reproduction and population growth until the resources are not so abundant anymore, and full and hearty competition resumes. There was plenty of competition in hunter gathering societies. For basic necessities and especially for the right to have children in an environment of relatively low carrying capacity.
If you do nothing about the murderer or rapist or general exploiter, then it might encourage more, and the exploiters will have gained the upper hand on your peaceful anarchist society. If you do something about it, then what you are doing is exercising a monopoly of violence by saying that no individual can harm another individual, but if one does, then everyone else is within their rights to harm the culprit. How is this different from police other than the fact that we have people who specialize in deterring criminals so that we can do other things? You might point to the injustice in our "justice" system and that police are often responsible for brutal beatings and whatnot, but I don't think that's the general tendency, and besides that, any system exercising a monopoly of violence is in danger of committing injustice. Say in our little hunter gathering tribe you find a pretty gemstone that I covet. I steal it from you, and you are outraged to find your gemstone has gone missing. Immediately fingers are pointed at the less likable, less smart, and lazy fellow who seems like he would steal something like that. He's injustly exiled while I caress my stolen bounty having gotten away with it.