Yaseen, The thyroid function tests you should request for hypothyroidism and Hashimoto's are: TSH, FT4, FT3, Thyroid Peroxidase antibody (TPOab) and thyroglobulin antibodies (TgAb). Unfortunately many labs decline to test FT3 unless TSH is suppressed <0.04.
Reverse T3 isn't done on the NHS as far as I'm aware and I'm not sure it's worth ordering private rT3 as there is often difficulty in interpreting the result.
Ferritin, vitaminD, B12 and folate are vitamin/mineral tests and are recommended as levels often become deficient/low prior to a hypothyroid diagnosis.
The cortisol, saliva, and synacthen are all tests for adrenals. Saliva tests aren't available on the NHS.
Testosterone is a sex hormone. Low levels aren't necessarily connected to thyroid but may be due to pituitary gland issues.
Chronic fatigue and fibromyalgia are syndromes which often resolve when thyroid medication is optimal but they are also illnesses in their own right and can be unconnected to thyroid. Rheumatologists test for fibromyalgia. I'm not sure which specialists test for chronic fatigue.
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