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.
Even if done out of selfishness, I believe altruism is selfless in nature.
Certain groups such as people who belong to churches or mosques are more likely to commit altruistic acts. The majority of people, anarchists included will not perform these actions that are beneficial to man as a whole on a daily basis. Self interest will triumph in the individual, however I do believe altruism can succeed if developed in a close group, such as a church.
Who is to say that we would work better in cooperation rather than competition? I think we have a larger sense of competition rather than cooperating with people. there will be those that wish to undermine the cooperation for their own personal gain, which creates a whole new sense of anarchy. I do like the thoughts of like minded individuals working in groups, which is sort of how communism got started. Living in "communes".
I feel like it is a good thought, but so was communism. Idealism in its purest form is a great thing, but nothing can be as simple and perfect as systems like this in my opinion.
I think that it could work but it could also just as easily fail. Theres no way to tell until somebody tries it, so I'm all for developing some ideas.
I admit I have no empirical evidence for my stance, but here is my reasoning: in order to compete, competing parties must spend resources on things other then direct progression towards their goal, thus reducing efficiency; while in cooperation, cooperating parties can spend all resources on direct progression towards their goal, thus acheiving maximum efficiency. Therefore, it is more efficient and effective to cooperate rather then compete. Regardless, here is an interesting read, as well as here. Note that neither represent my ideals or wishes, simply here as more food for thought. Also worthy of note: both of these were reactionary, meaning they arose in response to some opressive action or other. I do not think that's how anarchy will come about.