Anarchy does not mean that there is no structure, it means there is no hierarchy. I also think structure is necessary, which brings me to the first point you make.

Humans, in my opinion, are indeed selfish. However, the way I see it, anarchy is compatible with selfish people. The way I understand it (and practice it to the best of my ability) anarchy implies three basic rights or truths in regards to all people: freedom, equality and soliderity. Where it differs from other systems such as liberal democracy or social democracy is the third element: the idea that cooperation leads to better results then competition.

So in that sense, selfish individuals would thrive in anarchy just as much as altruistic ones; the mutual understanding that their cooperation would further them both more effectively then competition will be the force that keeps this system running. Of course, there is also the idea of free association, meaning that you are free to engage whichever individuals you wish, so long as they reciprocate and wish to engage with you. Thus we arrive at communities of like-minded individuals, which will sustain each other through the aforementioned understanding of solidarity, and the usage of technology...

But that's already straying into transhumanism and that's a whole other topic.