1. Obviously, she showed poor judgement, naivete and callousness. However, I think it was also accidental that the posting became public. If I remember correctly, she thought it was a private posting that was only supposed to be visible to specific friends
2. Do these things make her bad teacher? Not necessarily. Teachers constantly make similar comments in private to their colleagues and family. It's called venting. Teachers do not necessarily believe such comments. Rather, they are sometimes simply expressions of frustration. Most of us who have taught in a low income school or in one with poor disciplinary support from their administration has experienced periods of excessive disciplinary problems in the classroom and the frustration, anxiety and stress that go with it.
3. Should a person be stripped of their career and dumped into poverty for a mistake that did not directly harm anyone? Of course not. Get her some professional development to deal with the disciplinary problems, some tech support to train her to be more savvy about the internet, some mental health support to deal with her anger and anxieties.
Children are not more pure, innocent, worthy or deserving of rights and protections than adults by virtue of their age and status. Yet teachers ARE expected to be Madonnas, perfect and pure in every sense. We are not permitted the luxuries afforded other adults, like the right to get angry, yell, swear, pee when we need to, or to make mistakes. We are expected to accept low pay, demeaning treatment by politicians and the press, long and challenging working conditions, and to accept them with smiles, hugs and tenderness because we are seen more as protectors of the "innocents" than as workers trying to support our own families.
Let's also put this in perspective. The harm from firing the woman is clear and immediate. She is unlikely to be able to teach ever again. She is unemployed (probably without benefits) and thus forced into short-term (if not long-term) financial stress. It will be very difficult for her to get another job soon (let alone one with the security of teaching) due to the still high unemployment rates and the fact she had been fired for disciplinary reasons.
The harm to her students is hypothetical. We do not know that her comments reflected her actual perceptions of her students, let alone her actual physical treatment of them or her behavior in the classroom. And while Children's perceptions of their teachers are influenced by the gossip of peers and parents, they are far more influenced by their teachers' actual treatment of them