Should children be tested for food intolerance?

No. As opposed to true food allergy, food intolerance is not well understood or characterized. The best example is lactose intolerance. The mechanism causing food intolerance is not thought to be via the immune system, but often more of a toxicity, or like lactose intolerance, an enzyme deficiency. Other than eliminating and retrying the food, there are no tests, specifically no skin or blood tests.
No. If a child seems sensitive to some foods, testing will allow you to eliminate these foods. In many cases other sensitivities will develop w/ time, which means more elimination. It is like a dog chasing its tail. Best work on reducing sensitivities with homeopathy or another modality.
Depends. Food intolerance is a term that covers all adverse reactions to foods including food allergy, food poisoning, inability to digest a food, and even drug effects of a food (think of the stimulant effects of coffee or chocolate). An expert in food allergy can help you sort out which is the likely cause, perform testing when appropriate, and offer the best interventions or diet changes.
No. There has been little information to prove this point, you can however supplement their diet with a vit b complex tablet and omega 3 (fish oil capsules) research and current data indicates benefit in children particularly with fish oil capsules. May also consider screening for sensory integration disorder.