This sounds so like my DD, also six. I'm actually about to try and get the ball rolling with school for an assessment referral. I did mention my concerns with her teacher at parents' eve in November, when they raised the issue of her lack of concentration, extremely variable handwriting and inability to finish work. They said ADHD hadn't occurred to them (despite saying all of that!) but that my added information was helpful and that they would observe with that in mind.
So, yes with the constantly being on the go and non-stop talking. Also agree with you on the social skills, very confident to talk to people, actually quite astute with her questions, but whether she actually listens is another thing. Doesn't quite hug strangers, but will hug people she hasn't long met, parents of her classmates etc. Generally very physical, has always loved rough and tumble play etc. Runs, rather than walks, climbs anything that's remotely climbable, can't sit still, constantly flailing or fidgeting.
Was constantly on the go literally from being a baby, was always very alert. Sleep was then a struggle, she rolled over early (12 weeks) and would spend hours awake in her cot rolling over instead of napping.... I have a photo of her at five months having rolled over, pushed herself up and pulled down the camera of her monitor... At six months she was commando crawling... Tbh, it never really felt like I had a 'baby'...
Also talked very early. At her two-year check, the HV said to me, "You do realise that most two-year-olds don't talk like this, don't you?" And hasn't stopped talking since...
Also, on a daily basis, huge emotional outbursts over seemingly minor things, huge overreaction to clothing that it too tight, not fitting right, labels etc...