I can only address areas of familiarity on my part. What stands out to me is the doctor trying to fine-tune your thyroid by having you on 100 mcg of Synthroid 5 days a week and 88 the other two days. Yet the doctor is ignoring the fact that your FT4 is only at 30 % of its range, and your FT3 is at rock bottom of the range. FT3 largely regulates metabolism and is used by all the tissues of your body. FT4 is important as a source of FT3, by conversion.
When taking T4 med it is quite common to have inadequate conversion of T4 to T3 For good conversion you need for ferritin level (storage form of iron), to be at least 100. Your med dosage is not even enough to get your FT4 to mid-range, much less enough to get FT3 to a good level. I expect that the doctor is adjusting med dosage based on TSH levels, which doesn't work: the most important thing is how you feel. From the following list of symptoms typical of hypothyroidism, which ones do you have?
Increased sensitivity to cold
Constipation (need to use laxatives or fiber)
Dry skin (have to use skin creme)
Elevated blood cholesterol level
Muscle aches, tenderness and stiffness
Pain, stiffness or swelling in your joints
Heavier than normal or irregular menstrual periods
Slowed heart rate
A good thyroid doctor will adjust FT4 and FT3 levels as needed to relieve hypo symptoms, without going so far as to cause hyper symptoms. I think you need to add a source of T3 to your med dosage and increase it as needed to feel well. Since you are taking thyroid med, I am not sure why you are taking iodine. Also, if your cortisol test is correct in showing high cortisol, that would further indicate the need for your thyroid levels to be higher. Since cortisol offsets thyroid effects they need to be in relative balance.
Personally I would stop worrying about all the other issues that seem to be on your mind, and get your thyroid levels adequate. I think you will then be surprised at the effect on how you feel.
If you want to confirm what I say, please read a paper I co-authored. Here is a link.
Low carb diet can affect thyroid levels. Did you also test for the actual thyroid levels along with TSH, If so please post those results and their reference ranges shown on the lab report. The thyroid levels will be more indicative of your status than TSH. Even though doctors like to believe that TSH is a very sensitive indicator of thyroid status, it does not correlate well with either thyroid hormone Free T4 or Free T3.