I have an active boy who is three. Mine though is not *THAT* active.
My kid definently goes stir crazy in our apartment. We have NO outside place to play.
I started him in preschool across the street from our place, even though I was very worried about his willingness/ability to join them for group activities, sitting still, stuff like that.
He went for six days before the break and is now back in school this week. So far, his teacher says he is doing wonderful. (the other reason I put him in though is I have some concerns about his speech and comprehension of language, and it is a combined class for children with and without delays/concerns. His eval for school showed that my concerns aren't just paranoia--but it is the kind of thing where you have to get to know him to see it.)
And now, I think because of getting out for school and because DH takes him places on the weekends, and because of some developmental leap too I think, I have a child who will sit and happily work on his puzzles at the kitchen table----for sometimes an hour!
I'm not saying school, or even more outings, are going to be the solution to all your concerns. But
I worked in the preschool program that my son is now in at another school in our district and I can say that I think it is wonderful. They don't push academics, they have a lot of time for free play--yes some structured teacher-led activities, but also a lot of time for child-directed, more than at a lot of preschools I've seen.
I would suggest either looking into part-day preschool or trying more outings on your own to fun, indoor places--and getting outside as much as you can when weather allows. It's amazing what even a short time outside can do.
(If you think the separation might not go well, find somewhere that will be open to letting you stay until he's ready to handle it without you. He might surprise you--I went with my son the first day and left at the 'free play' time--he was so excited about that, he just hugged and kissed me, said 'bye' and that was the end of it! The second day, he started to cry but stopped when his teacher helped him find a puzzle. Hasn't cried since and now hugs and kisses me at his classroom door and I leave.)
I also think looking at diet is a great idea.
I second the total elimination, I have experience with sensitivities and breastfeeding, and it takes a good couple of weeks that way---and that is with it broken down by me and the little bit he was getting via milk! I would try it for a good six weeks to see if there is a change in behavior--the first month to see if it is totally gone, the other couple of weeks because it just might take it being COMPLETELY out of his system to see the change.
I've heard some remarkable stories of changes that happen by eliminating something from the diet.
then you could try adding back in very little and slowly anything you want to try---he may, like my son, be able to handle a little bit of ice cream or cheese sometimes, for example, but not drink milk. (my son's stuff isn't behavioral, it's digestive, but with elimination and gradual add-in, he can now handle some ice cream, cheese, yogurt, and dairy as an ingredient in cooking--and drinks goat's milk.) (I chose to try cheese, ice cream, etc just cause I couldn't imagine telling him he can NEVER have pizza, or ice cream at a party stuff like that.)