I will say this, Black_MiD, I admire your belief in humanity. But your society is based on assumption that has no scientific proof.
You say we are all naturally equal. This suggestion is easily refuted. Some people are born stronger than others, some people are born crippled. I am strongly near-sighted, other people are not. It's quite clear that humans are not naturally equal, and some are definitely stronger than others.
You say many of these desires by humans are a cause of our culture. First of all, that's an assumption, a nice one, but an assumption nonetheless. We know that humans didn't too well without a state, because more than 70,000 years ago, humanity was close to going extinct. The agricultural revolution was a major stepping stone for insurance the preservation of our species. But for societies to function on such a large scale that agricultural demanded, proto-states began to form.
You propose that humans are not greedy by nature, but that our current culture enforces it. Isn't it equally easy to suggest that our current culture enforces it, because it is human nature to be greedy?
But even if we assume that humans are naturally equal and these human conditions are created by our culture, most of the people in your new society will have been adults in our current society and their desires, beliefs and whatnot aren't going to go away, even with the education you speak of. A lot of people, particularly older people, are far more grounded in their beliefs and stubborn against change.
I admire your vision, but I just don't believe it will work with humans. Humans are too different. And as history has taught again and again, any society we create will always have someone to oppose it. Who is to say that someone won't come in and create a totalitarian regime somehow? Sure, the same thing could happen to our current systems. But it just seems easier when everything is voluntary. Anarchism and communism assumes the same thing; that every human is willing to go along with it. I don't believe they are. Regardless of the amount of education you give.