Cow's milk allergy is very common and usually disappears by age two to three years, but may occasionally be lifelong.

First eczema beginning at age 6 weeks to 3 months, then food allergies, then asthma usually before age 5 years, then environmental allergies.

From the Calgary Allergy Network webpage: Milk Allergy

This sounds an awful lot like some of what you've described.

A doctor at theinformedparent.com mentions briefly in a Milk Allergy article that:

Depending on the individual patient the symptoms can range from nasal discharge and cough to diarrhea, vomiting, rashes and behavior changes.

Does not go into what kids of behavior changes or anything, but, it sounds like you are on the right track.

Keep digging.

It was through my own research online that I figured out my DD's problem with her ankle was JRA, it took me months to convince DD's doctors to do the definitive tests for it.

Is milk, milk products, or foods possibly processed with milk still part of your child's diet? It could be a lifelong allergy to milk protein. You could try a complete elimination diet of all milk and milk products and see where that leads you, though you should probably research the best way to go about this before you begin. Forewarned and all. I did an elimination of wheat last year and four weeks later felt like a new woman. Now when I eat wheat my tendonitis acts up, my joints ache, I get brain fog...it's very obvious that I need to avoid it.

I suggest, if it is at all possible, doing some research on homeopathic constitutional remedies. If you're comfortable with it and have a practitioner nearby who works with children, honestly it would probably be your best bet for medical treatment aside from changing the diet. Naturopaths and Homeopaths tend to take food allergies and intolerances very seriously, whereas MDs often do not.
Good luck.

Anhata
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