Starchild:
Good questions here, and coming from someone I take to be an anarchist at heart :-). A few comments, without pretending to offer definitive answers.
Your concern about "rule of law" vs. "rule of men": I always liked the sound of that phrase, too, but I'm not sure what it gets us. Notice the degree to which our supposed system of rule of law is de facto a rule of men: Every law requires judgment and interpretation for its enforcement, for better or for worse. What we can easily end up with is the rule of men masquerading as the rule of law--arguably the worst arrangement. The idea of anarchy is to avoid either: _no rule_.
Your concern about bringing people to justice: In the Icelandic system (I recommend Jesse Byock's _Viking Age Iceland_, from Laissez Faire Books), no one was required, or forced, to participate in the justice system--to show up in court, or to abide by its dictates. But if you opt out, you make yourself a literal outlaw: You will not be able to enforce contracts against anyone else. That means you have essentially declared open season on yourself. In rare cases in Iceland, someone was simply banished from the island. No one forced them to leave, but they could perceive that it was in their self-interest.
Your concern (shared by our future mayor--thanks, Michael) about justice for the poor: Bringing someone to justice does cost money. In the Icelandic system, again, if you couldn't afford that, you could sell your claim to someone who did have the means. That way you at least got something in compensation. Under our present, punishment-based system, victims don't get anything at all, unless they have a tremendous capacity for schadenfreude.
Your concern about our current litigation explosion: That's something I would like to know more about myself. I assume there must have been some change in the legal environment in recent decades to bring this about, but I don't know what it is. Juries seem these days--but not earlier--to treat awards like lotteries. I think free-market arbitration agencies unlikely to produce such verdicts as the famous hot coffee one against McDonald's, which are very widely perceived as outrageous, and not just by potential defendants. Agencies would have a very hard time getting business from any potential corporate defendants after a decision like that. In general, if arbitration agencies are perceived as having a bias either for or against wealthy clients, they will have a hard time getting business except when the two parties perceive themselves to be about equally wealthy, which is a rather specialized market niche.