This is such a confusing argument! Shouldn't our goal then be to limit power instead of giving it to those who want it most? The police system attracts violent thugs who want to boss people around. The American government attracts greedy sons of bitches who want to get big fucking presents from their rich friends and make sure they stay rich. Capitalist business structures encourage a small group of enormous dicks to take all the money their employees generate. The system's current state is fucked up by the small group of assholes, and they've got us all convinced that's for the best! What you need is a system that actively works to keep power evenly distributed and democratic. You need a populace that is aware of the threats those selfish individuals pose and understand that they have the power collectively to stop that from happening. Yes, if everyone just sits on their butts and watches some motherfucker take over, it breaks down. But we know better now, so we can adjust things and make sure that doesn't happen. We just need to tear shit down first.
To eliminate poverty, one needs to examine how poverty began. In hunter-gatherer societies, there was no poverty. Poverty and, more importantly, the entire concept of class originated with the development of agriculture. Suddenly it became valuable to enclose and own land instead of simply living on it. Over thousands of years, this ownership became concentrated in fewer and fewer hands. Ownership = power. Ownership is limited, and therefore so is power. If power is limited, then those who seek it will also want others to have less - that's how they get more. Once people who want power have more than somebody else, they use it to increase their ownership relative to other people. Those power distinctions grow to create poverty among the powerless. What poverty really is is not poverty of money, but poverty of power and ownership. Therefore, in order to eliminate poverty, ownership (and thus power) must be redistributed to the people as evenly as possible.
As for your second question, people are certainly more capable of self-regulation than they are of regulating others. If you believe people are too shitty to be in charge of themselves, than they're certainly much less capable of being in charge of more than just themselves.
Ironically, you have a sub 12th-graders understanding of what anarchism actually is. Anarchism =/= no laws. Anarchism = as little hierarchy as possible. There would very much be law and order, but it would not come from on high.