I think you need to find out a little more before going in all guns blazing. If she's making a point of correcting mistakes constantly for one upmanship or to entertain the class then she should be pulled up on it. If she's being arrogant then the form teacher telling her so isn't necessarily inappropriate.
The same goes for a little "jibe" from a teacher. If she's doing her best to make them feel uncomfortable then a "why don't you ask Miss dictionary" isn't out of order and certainly not bullying. As she matures she'll learn that she'll be treated the way she treats others and this seems like a gentle introduction.
Of course, pointing out a mistake is fair enough but there are ways of doing it and I wonder if a 13 year old would always do so appropriately.
"She now comes home and tells me all the words the teacher has misspelled but doesn't mention it to the teacher"
Sounds like the teachers' methods have worked then.
FWIW, teachers rarely (maybe never) hate children. The children who think so are greatly over-estimating the effect that they have on their teacher. Rude, arrogant and annoying children are much more likely to attract ambivalence.
I worked well at school and did well. Head prefect, school colours etc and enjoyed it. For some reason though, was an absolute shit to the RE teacher. Never enough to get into trouble for it but I walked a fine line. It was like I saw that 90 minutes a week as a chance to rebel against the system before going away again to work hard in every other class.
Why do I feel that there's a bit of this going on here and that "every single teacher I saw at the last parent's evening gave glowing reports and said she was a joy to teach" is nearly always mentioned by parents when their offspring have a real issue with one teacher / aspect of the school.