It would seem that many objectivists spend so much time thinking about philosophy that you forget about the sciences that might help you achieve the kind of world you would want to live in.
It sounds like all arguments against Anarcho-capitalism from the objectivist point of view are more or less based on the assumption that having more than one protection agency would cause them to get violent with each other.
Rather than just depend on this assumption we should look into it in depth. Open, legitimate firms that put time and effort into maintaining a good reputation have many huge competitive advantages over underground gangs.
This is why the bootleggers of the 1920s fell from power so quickly after prohibition was lifted, because gangs are not competitive.
If there was only 3 competing defense firms, there would be the very real possibility of the type of chaos that I'm sure you all associate with anarchy. But if there are +100 firms, or even a lot less than that there is almost no risk of such a thing happening, because if a few start fighting, than they would rapidly lose power and resources(fighting is expensive) and a few other firms could step in and steal all their customers.
If the current performance of police departments is any indication than any private firm with more than 1000000 clients would be well above their optimum size. This means that in America, the number of defense firms would be a lot closer to 100 than three.
Each firm would have a slightly different understanding of the law, but this would not really matter. It is much more cost efficient for them to solve their differences through a simple bargaining process.
If you want a much more detailed description of the process I would suggest you read The Machinery of Freedom by noted economist David Friedman.