So two year old kids should be able to run out into the street and get themselves killed? But what if their parents (If parents even exist in your "society"

) stop them. Are they then breaking the law? But wait, there can't be any law, that would be "Authority"

And what if I decide to hold property by force of arms (Since there is certainly no such thing as legitimate property in your communitopia). What if I hire (Let's say I use gold, one of the admittedly good things about this society is that fiat paper money probably does not exist) to hire an army to do so? How are you going to stop me? Laws are a form of authority, and therefore I guess whoever has the most force wins, even though me using that force is also making me not an anarchist.
This literally makes no sense. I'm not sure what exactly you would call what Rothbard advocated but I would definitely say that what he proposed made a lot more sense than what you propose, although I am not exactly a Rothbardian.