My son is 6 too and a lot of what you’ve said resonates. I also think he might have ADHD.
I’ve never been contacted directly by school in the way you’ve had, but a couple of things have come up at parents’ evenings: shouting out, making silly noises randomly, not doing what he’s asked first time. He is also a bit of a pain at clubs, he fiddles, looks around, doesn’t look like he’s listening at all. Interestingly he’s fine at swimming, which he loves and is a small group of about 5 maximum. I asked his swimming instructor specifically as he’s been going a while and I was interested to hear that he listens very well there and knows exactly what’s expected of him.
He’s very fidgety, jiggles his legs when sitting listening to a story or playing a game. He concentrates on things that interest him like Lego, and to be fair he can get through school work, reading books, homework etc pretty well. He rushes work a lot, to get it done, so his handwriting isn’t the best, but I personally don’t think it’s awful either.
He’s like you describe your son with the instructions at home though. When it’s something he knows is part of his morning (for example he knows on school days he has to get dressed before heading downstairs) he’s fine, but if it’s a request like go and get a jumper and your toothbrush from upstairs, he either won’t, or will forget one, etc.
Interestingly he’s great on his bike and learned to ride, no stabilisers, age 4!
It’s difficult isn’t it. Some days I think my son is fine and he’s a lovely, kind boy so much of the time. He’s very sensitive though, also gets upset easily if he can’t do something, and I know his teacher thinks he can be disruptive and “challenging”.