Can I sue a teacher for writing thug life on my kids knuckles?

That's an outrageous act. I would seek discipline. Perhaps a nice Small Claims suit (for up to $10,000) would be worthwhile, too, for assault, battery, and intentional inflection of emotional distress. Again, that's outrageous; I am from a family of many teachers and 4 consecutive generations.

Yes. But also find out what your child was doing to result in such an extreme action. Your problem will be finding an attorney who is willing to take the case since you probably would not get much of an award against the teacher. Your more successful route will be to complain to the school principal and go up the ladder of authority until you find some help.

Anyone can sue anyone, but why would you?