Hi, you mention that talking to your daughter is like talking to a very aware and articulate adult. You also mention that she tries to avoid interaction with her peers, prefering adult company.
I wanted to focus on this, I have a son, who when young presented in a very similar way. Although he doesn't strop and is never violent. He is now 9.
Being bright doesn't cause social problems in itself, but for some children who are functioning cognitively well above their peers it can cause problems in socialising. Not through lack of social skills. A child may have excellent social skills, understanding how to behave in any given situation, for some bright children this means they prefer adult company. They may be confused, frustrated and bored by interaction with their peer group. I think this is perhaps more common with girls.
Emotionally, you mention that she likes to get her own way and that she reacts very strongly. I had this between 4 years and 6 yrs with my son. I eventually caught on, if you could ask him, he would say thank god I did. He argued with me from age 4, like a fairly well reasoned teenager! I spoke to NAGT who suggested that as he was both intellectually and conginitively very advanced he needed this acknowledged and needed to be treated very much as an equal. Albeit one with less life experience.
Talk to your daughter, how do her peers make her feel? How do her interaction with teachers and other adults make her feel? Does she want time to express her opinions and be listened to?
The NAGT are very helpful, they offer excellent advice and its fairly well established that some bright kids have social difficulties.