I have read your query in depth. Thanks for posting it.
Yes, we can always diagnose Hashimoto's thyroiditis
by specific antibody test like TPO (Thyroid Peroxidase
Antibody), an FNAC of the dominant thyroid nodule
can confirm the diagnosis of Hashimoto's.
You need to get the general allergy tests
done like serum IgE levels as a part of work up for the hives.
Please check your TSH, T3 and T4 levels now, if not done earlier.