I agree with many of the previous observations, ESPECIALLY having him evaluated for Sensory Processing Disorder. My daughter was diagnosed with it around age 2, and working with an occupational therapist was invaluable. It's a neurologic disorder. She gave us ideas for a sensory diet so that she could meet her sensory needs and be able to function better. So much of the descriptions you gave of your son sound so much like my DD- balance problems, rage, intensity, impatience, really strong food preferences... You mentioned that he was tested for allergies and had none, but food intolerance won't show up on a traditional allergy skin test because it's a different mechanism. My daughter had no reaction to nuts on a skin test, but when she eats any kind of nut or legume she gets mild diarrhea and her hyperactive behavior multiplies exponentially. It took me about 8 months to figure that out, but as soon as I pulled it from her diet the difference was noticeable.
Another huge help for us has been chiropractic. Before she started getting adjustments she didn't have a normal pain response. She could get seriously hurt on the playground, cry for 3 seconds, and then move on like nothing ever happened. Yet, try to put a brush through her hair and you would think that she was being stabbed. The chiropractor explained that while adjustments in adults are to correct chronic misalignment, in kids it's like a reboot for the nervous system. She stopped tripping over her feet, falling off of chairs (most of the time :-)) and is more consistently coordinated. And she can actually feel pain when she falls off the jungle gym.
Kids who are sensory seekers tend to act out to get sensory input in ways that aren't constructive because they are craving any kind of sensation, like hurling themselves into walls, running at full speed, climbing on anything and everything. If you can give him constructive activities that will meet those needs he may be able to function better the rest of the day.
The book "Raising Your Spirited Child" talks a lot about intensity and how to work with your intense child, and how to identify your own needs too so that your relationship isn't all about pushing each others buttons. I hope you are able to find someone to evaluate him and give you some tools to help him!