My son was a lot like your daughter. He doesn't have ADHD or ASD but was very high energy when he was younger, especially when excited. His sensory issues caused a need to shake his hands "to get some of the excess energy out."
He loved learning, especially science, and at six he didn't realize that it was rude to correct adults. He didn't think it was fair that it was perfectly acceptable for adults to correct kids but not for kids to correct adults when they were obviously wrong. After he finished kindergarten at a public school we homeschooled, more like unschooled, and I encouraged him to tell me if I said something that he thought was wrong and we would check it out together. He was usually right. Maybe this wasn't a good thing for my son socially at the time but I was not really worried about that.
I think he knew that not everyone else was as excited about learning things as he was but it was almost like he thought if he shared these interesting things with other people he could change their minds because how could anyone not think science was amazing.
My son loved to perform so we put him in musical theater at age 4. He loved it and it was a very good outlet for his energy and creativity. The older kids liked him for his sarcastic, quick-witted humor. He got to be with other gifted kids who didn't fit in but not all of them were gifted. I overheard a girl in his class tell her friend that he was speaking in "some geeky language she didn't understand." His friends understood and that was all that mattered.
His social ability improved over the years. His very social sister says he does not have any problems with social skills--at least with young adults. He doesn't share what he finds interesting unless the other person first expresses an interest. He continues to make jokes and he combines his acting ability with his quick-witted humor. I think putting my son in theater was the best thing I could have done for him.
Now that he is 14 some people describe him as quiet and studious, a big change from when he was six years old. I have also noticed that adults talk to him more--like people who stop to ask him lots of questions about the musical theater group, but it also happens a lot when we are shopping at music stores, Best Buy, video game stores, etc. I guess he looks both knowledgable and approachable. As he got older he learned to tone down his high level vocabulary and explain things in a way that I could understand often using humorous examples. He can't stand to watch videos of professors speaking in a monotone voice and slowly explaining things that should be interesting in such a way that would make most people lose interest in the subject.
