Your child sounds like a normal 4 year old kid. My son is 9 and is a fidgeter. My husband is too. He explained the fidgeting as a way to actually pay closer attention since you're meeting some kind of need to move. Kind of like how your computer has programs running in the background so that you can do your work on one tab kind of thing.
If your school doesn't have a gifted program, you probably don't have teachers trained to deal with gifted kids.
As for you not teaching because you won't be teaching CC curriculum, I would hope that any teacher worth their salt would point you toward some resources. Teacher supply stores likely have CC aligned curriculum throughout, as mine do. In fact, it's exceedingly hard to find curriculum in those stores that's not aligned, even in my state (VA) that is not a CC state at all. They are not worried about your teaching, so much as the potential of bad test scores because they don't think you can teach it as well as them or in the way they would.
