I would say that yes, having Celiac can cause other AI diseases to happen, as a result of prolonged inflammation and other genetic factors that may come into play. Ask anyone with multiples, like myself. I don't need a fancy study to figure that out.
I would say to try the gluten-free diet for awhile and see if your symptoms calm down. If they do not, and your tTg remains on the higher side for too long, then thyroid testing would be in order. As suggested, it would help to have your thyroid antibodies checked because that will tell you if thyroid disease is brewing.