Yes, it can be. People with autoimmune, often have compromised GI (Gastro Intestinal) system, and start to react to various foods. You need to take probiotics to restore gut flora (as well as prebiotic foods) The acid levels also matter. I would recommend looking into the bloodtype diet, which have helped many with these kinds of problems (including myself).
No I don't think there is a relation. Celiac would cause diarrehea, abdominal pain, and joint pain.
I have had an intolerance to tomatoes for the last few years. I was also diagnosed with Hashimoto's about 1.5 years ago. Is there a relation I don't know.
The only thing I know about is hashimoto people are more likely to have celiac. But that is a gluten intollerance.