I've only skimmed this but let me give a bunch of info that seems relevant. (As I understand it, the question is whether low thyroid levels could cause chronic fatigue and fibromyalgia). For background, I am a junior doctor with no specific experience in endocrinology and support healthy scepticism towards the medical establishment:

• Thyroid hormones are of course the first thing that one thinks of in people with low mood. Sometimes one will check the T4 and T3 as well as the TSH.
• it is also one of the first things that one would research and it looks like there are plenty of papers checking hormone levels in these conditions.
• People with hypothyroidism usually have high TSH. It would be surprising if this was different for people with fibromyalgia unless they had a specific insensitivity to thyroid hormone, which seems like a different hypothesis.
• people with hypothyroidism usually don't complain of pains in the same way as people with fibromyalgia.
• people with fibromyalgia usually have a high sensitivity to slight painful stimuli in a way that I've never seen from people with hypothyroidism.

I also must admit that the since the post seems to me to be not just wrong but also very slow-moving, I haven't read all of it. I would suggest rewriting this in a few paragraphs with citations to get feedback on a more brief version.