Mine is 8 and having those problems still in 2nd grade, in fact I thought you were describing her at first! She is 6 years older than her only sibling, so she was an only too for almost all her life. She isn't really like this at home either.
I don't have much advice, other than that the clinical psychologist we took her to suggested counseling with a gifted specialist who could help her learn to deal with her emotions. She also needs to go for social coaching. I'm surprised that your school, with all it's gifted experience, isn't more aware of these kinds of behaviors in gifted children or offering you more help.
