So if you piss off "the corporation", you have their private militia come after you. Then their courts sentence you to death, or slave labor. How is this anarchistic? Seems pretty top-down to me. Just because a company is running things doesn't mean it's not a government.

I just don't really understand libertarianism, I guess. It makes sense if you're filthy rich, but otherwise you'd just go from being middle class to a peasant with no rights almost instantly. Are all libertarians filthy rich and just very vocal (there seem to be a lot of them), delusional, misinformed, or am I missing some large part of the theory behind it?

The only way I could ever see the phasing of government out working successfully is in a shared economy (anarcho-communism). Not that I think human nature would ever permit a system like this until tens of thousands of years of psychological/spiritual evolution take place, but I think at the time humans are able to self-organize in every sense there will no longer be the need for personal possessions. Oddly enough, we would be a lot like ants.

Check your assumptions. Corporations are a legal fiction created by the State and would not even exist in a purely anarcho-capitalist society.

The intention of the law to "meet a need or solve a problem" doesn't necessarily reflect the actual behavior of the law. Adopting this mindset can also bring the attitude of "we need a law to protect children" while packing the "Protect Children Act" full of special interest material.

I would personally disagree with the assertion that the majority of laws solve a problem or result in a net meeting of needs.

This is very true, but it doesn't represent the views of all libertarians or even all anarcho-capitalists. For a much better approach to anarcho-capitalism—one that focuses primarily on economics and not on morality—check out David Friedman's writings and talks. Here's a basic introduction to his proposal of a polycentric legal system where the market produces all goods and services including law, law enforcement, arbitration (courts), and even national defense: http://www.youtube.com/watch?v=jTYkdEU_B4o. Along with Murray Rothbard's primarily moral/ethical/legal arguments (which seem slightly more prominent, and seem to be the ones you've encountered), Friedman's economic arguments make up the main "branches of anarcho-capitalism."

I don't know any libertarians or ancaps who insist that the non-aggression principle solves anything, much less everything. What we say is that it's simply the ethically Right Thing To Do, and that if "solving" a problem requires violating the NAF, then it's better left unsolved, even if there are negative consequences.

The idea that libertarians believe in some kind of "libertopia" is one of the most persistent, while completely unfounded and mistaken, myths that exists about libertarians. It's the libertarians who, more than any other group, take reality into account by acknowledging that we can't engineer a "perfect" society by using force against one another.

Libertarianism has pathologies at the edges? Big deal, so does every system that has ever been proposed as an alternative.

I doubt anyone reasonable even claims that a perfect society is possible - someone will always feel wronged regardless of their circumstance. I don't think it's unfair to say that the main function of a society is to reduce the likelihood that a person of said society gets wronged. Which brings the question of how does a society do that without enforcement of laws? If we accept the NAP is a law, who reprimands those who break it? Who actually defines what it is in the first place? Undoubtedly there are many different interpretations of it and it's not possible to achieve a feeling of fairness in a society if not everyone agrees on its basic structure and laws. If you feel like these are too basic of questions, feel free to link me to what you would consider a definitive explanation.

For me, it all basically reduces to the idea that Bastiat articulated, that "government is just the collective extension to our individual right to self defense". So even as a radical individualist anarchist / anarcho-capitalist, I can still see a role for something like "government", but only if it's use of force is restricted to defensive use, and it respects the idea that "if an individual doesn't have permission to do something, then multiple individuals don't have permission to either".

IOW, voluntarily organized collectives, assembled for mutual defense, OR individuals acting in self-defense, are how the NAP would be "enforced".

But again, there's no "perfect" political system, IMO. Conflicts could still arise, but I don't think institutionalizing the use of force is a good approach.