Would trespassing, stealing, lying, abuse, and killing increase to a significant degree under anarchism to justify having a "formal" government with elected leaders and such?
The things you mention happen regularly in the presence of a government. As well, from my experience, crimes such as theft are rarely punished. For instance, when a regular Joe has his house broken into and a few things stolen, I've never seen the cops roll out a "CSI" style investigation.
Do you suppose that there would be more "bad" people in an anarchist society? Personally, I think there may be a few more initially just due to the perceived opportunity to do so. However, I also believe that people tend to be a product of their upbringing. So, unless anarchism would significantly increase the number of bad parents, I doubt there would be more "bad" people roaming around.
The real reason (or at least a justifiable reason) why people are somewhat frightened of anarchy is because punishments become more harsh under such a system (or lack of a system) - banks don't get bailed out, regulatory capture doesn't exist, and horse thieves get hung. As with government punishment, sometimes the wrong people may suffer for things they did not do.