1) Diet is not important for the antibody testing. Some say that gluten will increase antibody levels, but I think that happens over time, and only for some. For the cortisol testing, I'd avoid caffeine as that will cause it to spike, so no coffee, chocolate, etc.
2) I'm not sure about those supplements in particular. Are you trying to diagnose Hashimoto's or just check antibody levels? The only thing I can think of that may be in your multi that would effect antibody levels in any significant way is selenium. Some people with Hashimoto's don't have elevated antibody levels. I'm not well versed on TH-1 dominance so I can't speak to that. They also wax and wane so you may not catch them high enough to diagnose. That doesn't necessarily mean you don't have it. I was finally diagnosed via ultrasound.
3) I don't think that hormone cycle is relevant to adrenal testing.
4) Follow the instructions on the adrenal kit. Try to avoid emotional or otherwise unusual stressers on the day of your cortisol test. If something happens that you think would cause your cortisol to be higher than normal, dump the tubes out and do it the next day.
I wouldn't listen to the person who told you that these tests are a waste. You definitely do need TSH, free T3, and free T4 tested as they're just as important, but these are necessary as well. The antibody test is obviously really important because many times lifestyle changes will help you, plus, you definitely want to know if you have an autoimmune issue. The adrenal testing is necessary because it's all connected and one being off will effect the other. Obviously, you already know this because you're doing the test. While STTM isn't the end all be all of thyroid info, it's super concise and helpful.