I'm still going through the full thread.
Firstly absolutely get your results (sorry if I've missed that you've done this).
Post them here.
I'd get in touch with the British thyroid foundation - they have a bit more 'official' welly over the other groups as they're linked to as they're the charitable wing of the British thyroid association, and are certainly not 'some group on the internet'. Ring them and discuss your experience.
I'd get the BMA book understanding thyroid disorders which clearly says keep tsh around 1. The ref range (check your results) goes as low as 0.3 (depends on your specific lab hence look at the range on your results) - so trying say a tsh of 0.6 for 3-6 months is not unreasonable. And really a gp should support this trial.
Tweak the dose to get this - alternate day 25/50's or 3 extra 25's a week. But you must stick to the same dose for a least 6 weeks to get the right picture on your results.
The book also lists medicines that interfere with thyroxine - sertraline being one of them. Also, avoid grapefruit as it changes the way the liver processes thyroxine (reduces). (Again, this book is not 'some group of the internet,' it's BMA.)
Tackle other levels e.g. Ferritin, vit d, b12. Get these in upper ranges. I definitely notice I'm better if my ferritin is over 70. I've change my diet to support this (no milk during the day to allow max absorption of iron from all my food - this worked for me but I might be weird).
And this is not woo - I can find a paper stating that this should be considered when looking at the bigger picture of a person with thyroid issues who is still symptomatic. There are some people who need t3 - it's relatively rare but this route must be followed to rule out these issues contributing to your symptoms before looking further into it.
Exercise definitely helps me once my levels are good. I've personally discovered I'm at my best just below 1. I thought I was well a bit lower but I wasn't maintaining muscle mass. Once tsh went up a bit and I did some yoga and physio to build muscle, I felt much better.
Both hyper and hypo states cause muscle weakness so I believe strength fitness for thyroid is really important- only possible when levels are good.
Apologies if I've repeated info, I'll try to read the rest. Small child here!