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So that got me thinking. In an AC utopia, is there anything to stop a parent from abusing their child? Since there wouldn't be any central authority, is that child doomed to a life of misery with little hope of freedom?

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First of all, AC does not claim to be utopian.

Children have the same rights to be free from aggression.

In state-dominated "utopia", what "stops" one from abusing their children? Obviously, the answer is "nothing", because child abuse happens despite the state.

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On a related note, what's to stop me from starting a new slave trade?

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Again, the same thing that would stop you in the state-dominated utopia. Nothing.

Of course, you'd still be liable for damages to the people you enslave.

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Also the whole role of a free-market judiciary confuses me. How is someone who aggresses against another made to pay for his crime? Where is he "tried"? Doesn't the fact that those seeking justice are also paying for a decision cause a big conflict of interest for the arbitrator? Even if there is a decision reached, how will it be enforced and by whom?

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Third-party, non-state arbitrators, paid for by the disputing parties, are currently used quite frequently in the US. Any arbitrator that merely issued judgements to the party willing to pay more would quickly lose any reputation, and their judgements would have zero value in the market.

On the flip side of that, reputable arbitrators will be able to issue judgements that carry a lot of weight with other reputable individuals and businesses. Most businesses will decline to do business with people that ignore reputable judgements. This means it will be difficult for such a person to move about (since roads will be privately owned) or trade with those that value their own reputations.