The best place to start is with the one page overview included in my journal. That also has a link to the UK site where the full paper is posted.
Yes, it does. Do you think you can get those tests done so you will know much more about your status.
Thyroid medication dosage should never be based on TSH levels. Our bodies evolved with a continuous low flow of thyroid med. When you take your daily dose all at once, it completely changes the equilibrium among TSH, Free T4 and Free T3. Most hypothyroid patients find that when their med dosage is adequate to relieve symptoms, their TSH is suppressed. That does not mean hyperthyroidism unless there are hyper symptoms due to excessive levels of Free T4 and Free T3. If you have been tested for Free T4 and Free T3, please post results and reference ranges shown on the lab report. If not you should do so and then make sure they always test for both every time you go in for tests.
Also, your thyroid status is determined by Tissue T3 Effect, which is a result of not only thyroid hormone levels but the response at the cellular level. One of the best indictors of tissue thyroid hormone levels is the Free T3 to Reverse T3 ratio. So I would also ask for a Reverse T3 test. The response at the cellular level is affected by a number of things, including cortisol, Vitamin D, and ferritin. So you should also get tested for those 3 as well. Since hypo patients are so frequently low in B12 that should be tested also.
A good thyroid doctor will treat a hypothyroid patient clinically by doing these tests and then prescribing thyroid med enough to raise your Free T4 to mid-range, and Free T3 into the upper third of its range, without being influenced by TSH levels. Once you get all those tests done then you also need to supplement as needed to optimize. D should be at least 50 ng/mL, B12 in the upper end of its range, and ferritin should be at least 100.
If you want to confirm what I have said, click on my name and then scroll down to my journal and read at least the one page overview of a paper on Diagnosis and Treatment of Hypothyroidism: A Patient's Perspective.