Sadly I think needed mh support with thyroid is v common :( but I does really affect so much of your life as you just can't function.
I don't believe bf has any impact; the lact consultant in my local lll group is hypothyroid and I've been assured there no difference. I believe bf actually affects your insulin growth factor (or what ever it is - the hormone that the 5:2 fasting diet triggers) so it's actually beneficial in a way,
No, I've been slim whole life and actually loose weight when very underactive, just puffing up a bit when slightly over. You actually loose lean muscle mass.
T4 is very stable as it has a long half life - naturally produced t4 does have a circadian rhythm as does Tsh. It's worth always getting your thyroid test at the same time of day for this reason. T3 does vary during the day; this is why it can be difficult to rely on as a test. An endo might test t3 to check for conversion issues (ie body not turning t4 into t3 effectively) but really should check other factors eg that Tsh is low (around 1), iron (ferritin) levels, b12, vit d, folate. B12 can be an issue as pernicious anaemia is another autoimmune issue that can be linked. But again you won't feel too great if it's low.
I was told by an endo (and prof) to get tests in the morning, not to take that days thyroxine till afterwards. To also keep a record of Tsh and any other results and how you feel.
I have to say, in a way, telling others my experiences and how to look after yourself helps a bit to make up for the sadness I experienced when things were not well addressed.
I have found the British thyroid foundation (charity) very helpful too.
Their website alerted me to the fact that a brand of thyroxine was faulty and being withdrawn (TEVA) - I was pregnant at the time and had a difficult time with my levels, also triggered anxiety as I knew it was important to have good levels in the first trimester - they weren't due to the brand, it was too weak. Then I switched and went too high.
If you are planning a second child I would defiantly ask to be referred to an endo, or at the least make sure gp follows the nice guidelines. I had to fight a bit to get my gps to realise you had to up your dose.