A brief comment. You said you don't live in an abusive household but yet you advised your son to tell a girl that he felt bullied by to threaten to break her nose. Sure sounds abusive to me. I suggest that if this is an indication of how you solve relationship issues, she is following your example. She is having difficulty with relationships and needs to learn non-violent ways of handling them.
I suggest she's lacking in social skills. I can't begin to know what she needs to learn but I do agree with you that beating up on people does not solve anything.
My daugher's and granddaughter's school has a friendship group run by the school counselor. I suggest calling the school to see how they can help.
I think I understood from your post about bullies that you don't think the school can be helpful but I have found, thru experience with a number of schools that they can be helpful and urge you to give that a try.
If that isn't helpful, I urge you to get her into counseling. I also urge you, in love, to take a close look at what you may be unwittingly teaching your children. I had anger issues for many years without realizing they were negatively affecting my friendships. Counseling helped me tremendously.
You should be sympathetic with fat people because they are people. Being fat is not a simple matter of eating less, even. No one is perfect. We are all people who need love and support. You took offense because you felt that some were critical of you. Your feelings towards those who didn't seem sympathetic to you are an illustration of how you're not sympathetic with fat people.
"Love does make the world go around." And "what goes around comes around." When we aren't sympathetic with others, others have difficulty being sympathetic with us. We create the environment in which we live. I hope that you will be able to teach your children to "love one another" while you're home schooling them. We are all human, make mistakes, but are still loveable in some way.
Later: If physical discipline works, why is your daughter acting like a bully? You're right, it's none of our business, how you discipline but you are asking for ideas and how you discipline does affect how your children behave. Spanking can be effective when administered in a calm and non-judgmental and loving way. However, the tone of this post and the one about someone bullying your son, sound quite angry. If you're angry when you physically discipline your kids, I suggest that they equate your discipline with appropriate behavior when angry. I'm sure that is not your intention.
I'm sure you feel defensive as you read these posts. I urge you to stop and consider what is actually happening in your home, without being judgmental of yourself and anyone else. Be as neutral as you can be and observe yourself and your children.