Basically, thyroid manufactures mostly T4 and a little T3. Some of each is bound to protein, while the rest is free to participate in metabolic reactions. Thus, you can measure TT4 (total T4, which is bound T4 + free T4), FT4, TT3, and FT3.
In tissues, T4 converts to T3 as needed. Problems arise if genetically you have difficulty converting T4 to T3 or if you are ill. Then the body converts T4 to RT3 (reverse T3), which effectively blocks T3 from its own receptors. In that case you could have a good level of FT3, but it will be blocked by RT3 and you'll still feel bad.
FT3 becomes important as a driver for the speed at which your metabolism works. It can also be evaluated with temperatures, where a good morning oral temperature (after 10 minutes with a manual thermometer such as Geratherm) before getting out of bed is 36.6-36.8C (97.8-98.2F). During the afternoon it's nice if a temp reaches 37C (98.6F) or a bit more.
TSH is a pituitary hormone. The pituitary senses the level of thyroid hormone passing through it in blood, and if it doesn't sense enough, it starts the process of raising thyroid hormone production by raising TSH (thyroid-stimulating hormone). Unfortunately, the pituitary has no idea what tissue levels of T3 are, and tissue levels are key to feeling well. It's possible in the scenario of good levels of FT3 but high RT3 that TSH would be lower in the range, because pituitary would be happy with the level of FT3 in serum, even when your temps and tissue levels of T3 are bad!
Tissue levels of T3 can be influenced by iron, adrenal health, B12, and sodium and potassium, to name a few. So, yes, if you can swing it, test serum iron, saturation, and ferritin. Be sure you take no supplemental iron for 5 days, and always test fasting early in the day, such as about 8 a.m.
I need to take 8000 iu Vitamin D in order to maintain my levels, so before you start lowering supplementation, be sure you need to.