Re: What tests to ask for??
TSH is generally useless. In the rare instances where it rises out of range, it's the only 'proof' most MDs will accept. The problem is, too often it doesn't rise out of range for decades after the thyroid hormones have become deficient.
The absolute minimum tests needed for screening are TSH and free T4. If you have multiple hypothyroid-like symptoms, thyroid antibodies tests (both TPO and Tg) and free T3 would be very useful. Even if both antibodies tests come back negative, that isn't proof that you don't have Hashimoto's; it might only mean the test could be one of the up-to-20% that are false-negative.
That's it. The tests you don't want are total T4 (or "plain" T4), total T3, T3 uptake, or thyroxine free index (aka "T7"). None of these are useful. Don't accept them as substitute for genuine free T4/3 tests.
"We can complain because rose bushes have thorns, or rejoice because thorn bushes have roses." Abraham Lincoln