You do not need reverse T3, no. That's a very expensive test that doesn't give you any useful information. The basics are:
TSH
FT4
FT3
TPO antibodies
Tg antibodies
vit D
vit B12
folate
ferritin
Get those tests first, and see if the results suggest the need for further testing. No point in testing for Grave's if the above come back showing hypo. But, always a good idea to test the Hashi's antibodies - TPOab and TgAB. The symptoms you suggest could be due to a 'hyper' phase of Hashi's, or just hypo symptoms, as many of them cross over.
I'm afraid I don't know anything about Lyme or testing for it.
They will show if you're hypo, hyper or euthyroid.
If you have Hashi's, it will probably show up on the antibody tests. But, might not. Antibodies fluctuate all the time, so they could be low when tested. Up to at least three antibody tests are needed to catch the antibodies when they're high. But, even if they are never over-range, you could still have Hashi's. About 20% of Hashi's people never have high antibodies. So, it's really a case of not being able to prove a negative.
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