Just a few thoughts thoughts for you, not sure if any of it will help or be relevant. I have also had a shed load of tests done in the last year to try to find answers. I am currently diagnosed with hashi's, fibromyalgia and lactose intolerance. I still am not 'well' and just want to feel better. Anyway.......
You have hashimoto's and are on thyroxine, but how often do you have your levels checked? And does your doc test for FT3 and FT4 as well as TSH? You can still have symptoms if your meds aren't enough for you, I know I did and ended up having an increase in dose.
Also, it is possible you have fibromyalgia. It seems to be a bit related to hypothyroidism. And you get 'flares' of it like you describe. Lots of pain, extreme fatigue etc. I was told I have it after the thyroid meds did not alleviate my symptoms. Might be worth trying to get your doc to give you a referral to a rhematologist? Not sure how helpful your doc is but if they are not willing to help you I suggest you try elsewhere if at all possible.
As for feeling great on gluten free, did you/do you have tummy troubles? Not sure if this relates to you but I always thought I had IBS made worse by bread/pasta etc for over 15 years, but my doc recently sent me for hydrogen breath testing. Came out that I was lactose intolerant. Lactose is in a lot of foods, not just dairy. (I am not one to drink/eat a lot of dairy so never put my tummy troubles down to it) The tests are not expensive, I paid AUS$55, but they are not rebateable by medicare.
Not sure about most of your tests, but I'm sure that even us females need some testosterone don't we? I would check again with your doc.
Sorry you are feeling so bad and hope that this bad spell passes. It is very frustrating not getting answers and I get that you feel you will never be normal. Best of luck and keep pushing. Please keep us posted on how you go.