The way I understand it, our noodles( brains) are mostly made of fatty tissue and nerves. So ifn' that be true, it makes sense that if you can't absorb fats correctly, or at all, that your noodle might be affected. So, it seems like celiac could definitely cause mental symptoms. Not to mention there is a lot of research showing it does. Try a saerach on celiac and pschisophrenia or something spelled similar to that, and celiac and gluten ataxia or celiac and depression. There are lots of articles about how gluten can affect the brain and personality.
There is a possibility you are reacting to another food also. It is not unusual for celiacs to develop additional food intolerances beyond gluten IMHO. Check out some of the member signatures to see what I mean. If it is another food intoelrance (check for gluten first of course) then an eliminaiton diet is a good way to proceed.
Yes it could be dairy products. The casein can be a big problem for a lot of people plus there is a ton of hormones in dairy.
Here is a list of the bad things in commercial milk.