I supressed the island part to focus on the first part of your argument wich is: in capitalism you're free to pick and choose and that capitalism is not explotative. Of course, alone on a island there's no capitalist system because there's no one to trade or sell things, the space would be only mine.

Exactly, there's no economic system there, and yet the man suffers from exactly the "oppression" you describe to a much larger degree than someone living under capitalism.

Lol, I don't want people to serve me!

But you do. You think that you working to feed and house yourself is exploitative. The only solution to this problem is for other people to give up their property to you, property they worked for, property they homesteaded. Also, you're not being submissive by taking a job. When you take a job it means that you'd rather use the time and effort to perform that job than anything else you could have done with that time and effort. The same is true of the employer. He'd rather use the money and resources to hire you than anything else he could have done with that money.

But following your response to my question about how the State would vanish: what would make these enormous corporations crumble if they have the money? Not having the State would not eliminate their power.

They would be faced with competition, global competition. They wouldn't be protected by the state anymore. I don't see how they could survive. Having money doesn't mean that you're guaranteed profits.

Let's say there's an anarchist paradise and there are two communities. One is AnCap and the other is AnComm. AnCap keeps capitalism going at full speed, soon their resources run out, AnComm has resources they need. How would the situation be handled?

Economists describe the fate of communally owned resources as a “tragedy of the commons,” after a famous article by Garrett Hardin. In Hardin’s original historical example, before the great enclosures of pasture lands, herders would systematically allow their animals to overgraze, i.e. to eat more grass than would allow the pasture to sustain itself. In modern times, communal lakes and streams are plagued by overfishing and not enough fish are left in the water to sustain the population. Everyone is aware of the problem, but no one has the incentive to change; even if an individual fisherman limits his catch, that won’t prevent the next one from taking the fish himself. The way to solve the tragedy of the commons is to convert the public resource into private property. With privately owned and managed bodies of water, overfishing would be as obsolete as overgrazing.