There are two distinct parts to this topic:
Note that FB has usage rights and they could censor stuff, and they do. They block terrorist propaganda, for example. The question here is should it be an all or nothing, or as-decided-by-FB? Regarding the lawsuits, note that in the U.S. there are number of protections on media reporting of minors (as an example). If FB allows such reporting unfiltered, can they be held accountable? I would hope not, but it depends on what the court would agree.
It isn't easy being the parent of a six-year-old. However, it's a pretty small price to pay for having somebody around the house who understands computers.