In answer to your 3 questions, I would say absolutely yes.
We all learn in different ways....some auditory, some visual, some kinesthetic..most of us have a dominant learning system.
But what is sometimes overlooked is that we also have an optimum level of arousal. Most people wont learn well if they have had a heavy meal or run 5k. Most will learn better if they then go for a walk or do some deep breathing and stretching.
So for kids, they not only need different methods of teaching the material, they also need different methods to 'prime the pump' so to speak.
Some will need a chew toy/gum, some will need to wriggle, some will need to click their fingers or chew their nails, some will need LOTS of movement before they can concentrate.
Sensory aids would be appropriate for them (like blu-tack, gum, mints, wriggle cushions, movement breaks). And for most kids, they will need the same sensory aid for homework or other time they need to concentrate.
Yes the teacher should be able to give these out. IME most teachers know the children who 'need' an extra something but they are not always confident about making a call on it and prefer an outside 'expert' to give specific advice.
Yes the child is normal.
How does your engine run' is a good programme used to teach parents, teachers and kids about this.