If I told you that the teacher had to draw the classroom curtains to stop my son gawping out the window in year 3 would it make you feel better? When we asked what he always stared at, he told us he was looking at the trees, wishing he was climbing them.
He also once reached the end of lessons with blank pages, as he missed instructions, finally his chums special needs teacher took him under her wing too, and helped him focus a bit more.
Year 4 saw him turn himself around, he had a cool, young male teacher, but this year has all got a bit too much, and he is back to day dreaming. He then gets upset when he realises he has run out of time, and has "done rubbish work". To be honest it is all the " boring evaluations" that are turning him off, he hates them, plus the increased volume of writing now required.
I don't want to sound patronising, because I am no expert, but I am a firm believer that children, like dogs, need to be run. I am sure all this wet play of late has not helped, they seem to concentrate so much more when exercised in the fresh air. This is half the reason I let mine do so much sport and spend so much time in the woods, because they are far less irritating and far more focussed afterwards.
Ds1 is our Mr fidget, he cannot even have a rubber or spare pencil on a desk when doing a test, because he spends longer fiddling with them than time allows. He also used to fall off chairs like your dd and is never still, always jiggling about, I cannot stand sharing a sofa with him, it makes me feel seasick!
I don't think it has any label other than childhood to be honest, irritating as it is to us completely rational adults (says she with very sore picked at cuticles) they just need a poke with a cattle prod sometimes.