In general, I think that most of these concerns are based on the assumption that all government would be abolished at once. If that were to happen, I think that would be more indicative of some sort of catastrophe and not a peaceful transition. In that case, dealing with the catastrophe itself, and its side effects, would probably be terrible all around, but not a cost of anarchy, but a cost of relying upon a long chain of other people for basic needs.
To address your first concern, about the loss of law enforcement as a deterrent to crime, sure, some people probably would take advantage of a situation where other people were completely unorganized with no means of protecting themselves. However, thinking of police as the only means of protection, and that security, as a concept, is only possible with a government, is silly.
I think one of the major hidden costs of a peaceful transition would be economic upheaval. Without a government propping up and defending the major moneyed interests, they will collapse. In the short term, that will cause a large deal of unemployment, foreclosures, scarcity of resources, inability to distribute available resources, and other related problems. In the long term, those problems will be dealt with because the barriers preventing people from undertaking those tasks will be gone as well. However, there is a large potential for harm that must be addressed to prevent cities from collapsing into diseased, starving battle grounds.
Another hidden cost will be losing a large degree of luxury that we currently take for granted. Our economy, like every empire, is based on slavery and military domination. Without slaves across the world to manufacture our goods, and resources stolen from other people, we will have to "tighten our belts" and learn to live with buying fewer fancy gadgets, exotic foods, and new clothes. I believe that we will be able to return to a comparable availability of goods over time, but the short term consequences of morality will be difficult.
I'll address your second question, about opposition, later.