It is difficult to recommend when not really knowing how you feel and what your blood tests look like. Hypothyroidism is such a general category.
Diet: I would say a PHD with more fruit. If you want tweak it, try even more fruit, more diary and less vegetables and starch. Do not eat massive amount of muscle meats. Broth is good. As cliff said - you have to get enough carbs, and not all carbs are equal. Some people, myself included, do better with fruit. And yes, as pointed out, stay clear of anything in the cabbage family. Do not take it fermented either. Peat style eating helped me.
Sleep/lifestyle: sure, if you are not dialed in with sleep, just do it. Go to bed early, sleep in a dark room etc. Chris Kresser's site as recommended by others is good for lifestyle issues and hypothyroidism, but do not think that simply applying the perfect lifestyle will save you.
Vitamin D - depends on your Vitamin D status. Get it tested. If under 40, you probably want to try supplements in addition to sun exposure. If very low, you should probably take 5000 IU per day.
Too much or too little selenium and iodine can be a problem. Try to eat fish, shellfish and mussels/oysters. If you take iodine, make sure you also take selenium. My personal experience is to take it easy on the supplements. Better to get a really nutrient dense diet.
Would in any case get a full blood panel to see what is going on.
Exercise - depends on how messed up you are metabolically and what you enjoy doing. Some strength training a couple of times a week should be good. Maybe some Sprint / Peak 8 (30 second sprints on a bike, in the pool or whatever, and then 90 seconds easy, up to 8 times). There is no one way. I am hypothyroid and I do short and mid distance triathlons. I am not saying it is smart, but it is my choice and after getting my diet dialed in, I am okay. On a regular Paleo diet with tons sweet potatoes, taro etc, I was crushed and had to skip training most days. I now always take enough carbohydrates before and after exercise, along with some quality protein. More specifically, I drink goats milk and orange juice before and after training. I eat something after. Works for me.
Lost hair - yes, definitely check out http://www.dannyroddy.com/