You say you believe in "the human capacity to resolve minor civil disputes"
Ok but what about crimes? What would happen to all criminals in an anarchist society?
That's a really dumb question. The term criminal is a social construct maintained arbitrarily by a self imposed powerful minority.
Explaining social constructions to statists seems to be a wasted endeavor, but as is.
You can't do almost anything against those who commit crimes.
"Crimes" in the technical sense of the word do not exist, without an established authority to label them as such.
There is however actions that human beings consider undesirable out of their own principle. There are many ways society can solves petty disputes such as theft, infringement, etcetera.
There are inner community debates on the capacity for violence within anarchy, but I personally believe in a strict adherence to non-aggression. That is different than pacifism, in that I believe others have the right to act in defense.