I think some of your symptoms are consistent with celiac disease, others may not be. The brain fog can not be explained with your age as you are very young. The skin rash could conceivably be dermatitis herpetiformis as it manifests differently sometimes, that condition sometimes takes a very long time to respond to a gluten free diet.
The eye issues are not typical of celiac disease. Thyroid issues sometimes manifest with eye related issues.
I am skeptical about your mother having a genetic gluten intolerance. The only genetic condition I have ever heard of that involves gluten is celiac disease, and it is not necessary for both parents to have the genetic code for celiac disease in order for a child to get it. Sometimes doctors will refer to a sub-clinical manifestation of celiac disease as gluten intolerance. At other times people who simply do not tolerate gluten well are diagnosed correctly as being gluten intolerant, but this being genetic is news to me, I'm deeply skeptical about it.
You should know that blood tests for the antigens involved with celiac disease will not be reliable if the patient has not been eating gluten steadily for a matter of weeks.
The gold standard for diagnosing celiac disease is still a biopsy of small intestine villi obtained via an endoscopy.