She sounds a lot like the 5 year old son of a friend of mine - turns out he is gifted (can read at a 5th grade level) but lags behind with emotional/social development. They did have him evaluated and the therapist said that this is very common in gifted children - they may be exceptionally intelligent in some areas (reading, math, etc.) but still be very emotionally/socially immature. He gets frustrated easily too when he doesn't get his way, when rules don't make sense to him but he still has to follow them, and he gets bored easily as well - which all makes him prone to acting out and having melt-downs, even at almost 6 years old. And stubborn doesn't begin to describe it - he has been sent to his room for time-outs for not listening and has screamed his head off for an hour straight because he doesn't want to be in there.
Because of this, I would agree with those that are saying it would be best to talk to your pediatrician about this and get a referral or recommendation for a child psychiatrist or other therapist evaluation. I hate "labeling" kids, but anything like giftedness, ADHD, ODD, Asperger's, etc. can result in what you are seeing and then it's a matter of determining what kind of behavioral therapy will help and having a plan in place. My friend's son will be starting kindergarten soon and right away they have discussed having an IEP in place for him and having him do reading, etc. with a higher-grade class as part of it.