Some. Some of the rare types are, but for the most part, the autoimmune thyroid diseases are not so much inherited, but the tendency toward developing thyroiditis is. So, if you have a parent or sibling with autoimmune thyroiditis, your risk for developing autoimmune thyroid disease is higher than that of the general population, but you don't actually inherit the thyroid disease.
They can be. Most thyroid problems in the US are caused by auto-immunity, where the immune system misclassifies one or more thyroid proteins as foreign (not belong to that person) and makes antibodies that target the thyroid protein(s). Auto-immune thyroid disease often does run in families, although family member may have different manifestations - goiter, nodules, too little or too much thyroid hormone.