Sensei is a total psychopath. Enjoys torture and murder. Has zero awareness of other people as people. Zero empathy. However, she only seems to target malicious scum. Except for the brother?
For the brother, we don't know what she did. Even if she intentionally pushed him down the stairs, broken neck would only be a small chance. Much more likely would be sprains, other broken bones etc. As for brother's characters, he did seems fairly innocent, just teasing/annoying a girl that he liked. But... Doing that a few times is one thing. Doing it every damned day? That suggests that he himself was uncaring of her actual feelings. And he was playing around with a switch [butterfly] knife for fun. Not a pyschopath, but not that innocent either. En route to being a thug.
(There is the question of what the brother saw young sensei do... Probably something like torture or live dissection of an animal.)
As for Akizuki... Also not a pyschopath. But also a participant in rape, torture, social blackmail. Loved his brother, but he's still malicious scum.
Sensei Evaluation... The sort of person where you're glad they're on your side. And where you buy a gun and start practicing self defense in fear of the day that she's not on your side.
Higher ups in the educational system... Looks like we've got ourselves a "problem solver"! Hmm... What other schools should we send her to once she's "problem solved" this school?