It really depends, but generally speaking, no. If the bullying is illegal, it should be reported to the police if the school is the first to know of it. The school my kids go to "loan" them laptops for each school year and we were specifically told that they (the school) MUST make sure the kids don't do the "cyber bully" thing on the school laptops as the school is liable. But under normal circumstances, I don't see how the school could possibly be expected to provide any consistent protection from cyber bullying off of school grounds.
That sounds about right. I'd add school-sponsored trips, contests, etc.
We cannot expect the state to act as parents-in-absentia. If they're using school property, or on chaperoned school time, then that's one thing, but generally the school cannot be held responsible for, and should not take part in, persecuting or prosecuting students for any actions or speech.