Orange juice. I've still not figured out all the other details as I've not seen any need to, and so don't endorse them, but as a starting point you can certainly do worse than look into Ray Peat's work since it has helped others in your situation. There's also plenty of questions on here if you search discussing various aspects.

To answer your specific questions as best I can in general terms. Sleep is always important, and if you're regularly up til midnight you're either young or really short-changing your health. Plenty of time outdoors during the day, but also plenty in bed at night.

You should be able to rocketship your way through exciting workouts, though where exactly that falls in the recovery from hypothroidism I don't know. You may also want to investigate Danny Roddy for more hair tales. There's so many things you can try, but without knowing the details of what you're eating etc. at the moment it's particularly difficult to determine the best course to start on. I guess the key thing is don't be afraid of experimenting.

Get enough carbs/protein, eat quality protein(balance muscle meat with gelatin) and get your carbohydrates from fruits preferably.

Eat more carbs. Please eat more carbs.

Can I ask why you are medication free? And how long you have been paleo?

I have been on thyroid medication for over ten years and they work well for me but I understand your reluctance to take meds.

To be honest regarding diet, I belive that the paleo diet should be a good fit for thyroid disease because most things normally recommended for thyroid disease are low carb, no soya etc. However bare in mind that these are the same diets that advocate eating grains and avoiding fat.

There is also an emphasis on avoiding certain foods such as cabbage, broccoli, rutabaga, cauliflower, kale, Brussels sprouts, watercress, and peanuts because of 'goitrogens which block the production of thyroid hormone.'

I am not sure if there is eveidence of these claims and I will google this after writing this but if anyone has a link to research showing that cabbage family veggies cause thyroid problems please add a link.

added: this should tell you alot : http://www.askthelowcarbexperts.com/2012/03/10-all-things-thyroid-thyroid-101-chris-kresser/

You may want to look through some of the thyroid information on Chris Kresser's site.

http://chriskresser.com/thyroid

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.

Anyway,

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/

Diet: This can be very individual in regards to macronutrient ratios. Trial and error can be the best way to figure this out for yourself. What is more universal is the need for nutrient dense food. Liver, grass-fed butter, wild seafood, oysters, local organic colorful vegetables (ripened in the field and eaten before they lose the nutrients,) and plenty of leafy greens.

Diet part 2: Avoid goitrogens like soy and anything in the cabbage family!

Sprinting and weightlifting: A good thing. Cronic cardio might be too stressful, but sprinting and lifting should help metabolic health. Pay attention to how you feel, and start slowly if you are nervous about it.

Sleep: Very important. Keep it consistent and as close as you can to the schedule of the sun. Allow your bodies natural hormone cycle to work. If it is really off you could try melatonin at night to teach your body to fall asleep before midnight.

Hair: What worked for me is externally applied iodine. Half a dropper of lugols 2% in enough water to soak my scalp, applied directly to the with a bottle with a dropper. Not the original lugols dropper, keep that one clean. Of course hair loss is complex, biotin seems to work for some, going no-poo helps some. Make sure to try only one thing at a time, give it a good lead time to see if it works, and the beast measure is probably how much is in your comb, not on your head. You will notice that you are sheading less before you notice that you are growing more.

Other: Look into upping the amount of coconut oil in your diet, some people notice that it works for hypothyroidism. Also look into iodine. Works really well for some, almost a miracle cure, makes some worse. Do your research and decide for yourself. There is a really long and growing thread on Marks Daily Apple Forums about it, and lots of other info around the Internet.