Samuel - I think it is remarkable that you are this interested in your health at your age. While I mostly agree with all the other posters about you not eating enough and not getting enough carbs, I want to share with you some things I stumbled upon that improved a life long battle with brain fog and inexplicable exhaustion.
Brain fog:
Do you really struggle with waking up by find it very hard to fall asleep at night? Like, you are tired all day but suddenly become more mentally alert around the time you should be falling asleep? Do you feel disconnected from your body and sort of like you are "watching" yourself? Do you have trouble concentrating on things you aren't interested in, but become completely absorbed by things you find rewarding? I know the answer to that last question is that everyone feels that way, but I mean is it very pronounced in you? Like, is it torture to do homework for a subject you hate, but you can lose HOURS on the internet researching a subject you like?
I ask the above questions because I went through this stuff for most of my young adult life until I was diagnosed ADHD. I never thought I was ADHD because I could always sit down and read for hours and I thought people with ADHD couldn't do that. But, I would start reading a book at night and stay awake all night without sleeping to finish the book - I remember starting to do that when I was nine. It is a common misconception that ADHD is the lack of focus. Really, it is just that you have SO MUCH focus on certain things and you can't apportion your focus appropriately. So, you'll get absorbed in a ticking noise but miss that math lecture completely. Or, get absorbed in a book, but miss that sleep thing entirely.
I don't medicate my ADHD. I did at one point and it dramatically improved my symptoms and calmed me down overall, but I am reluctant to advise anyone to take those drugs - especially a teenaged guy whose brain is still developing. They're a low level methamphetamine and no matter what big pharma tells us, we can't know the very long term effect of those drugs on brain development and personality. I will say that learning more about it and how to naturally address it will definitely help. My first recommendation would be to eliminate grains and dairy, but it sounds like you're on top of that! If any of this sounds like you, write back and I will write more about my coping tactics.
If you have gotten this far in this long winded post, maybe you don't have ADHD....
Exhaustion -
EDIT: Sorry Samuel! I misread your original post. You said it is WORSE in the winter. Ignore this part, then. I'll keep it in just in case it helps someone else, but I don't think it applies to you.
Just a stab in the dark, but maybe the sun is taking it out of you? You said that it was worse in the summer time. I think the majority of people benefit from the sun. But, not everybody. I have a really rare autoimmune disorder called Tumid Lupus. It means I am pretty much "allergic" to the sun. Even very low levels of sun give me a rash and wipes me out. IT SUCKS. But, not the end of the world. I was exhausted all through high school, when I should have been feeling my healthiest. I wasn't fat, but I wasn't healthy. All during that time I was eating a ton of grains, my sleep schedule was screwed up because of the ADHD thing, and I was worshipping the sun. Wish someone had told me then...
Going primal solved a lot of my energy problems, but the best steps for me involved avoiding the sun, eliminating grains (especially beer) and dairy, forcing myself awake earlier so I would go to sleep, not letting myself read at night, exercising when I don't feel like it, and using different coping tactics for ADHD.
All that being said .... you know your body. If something feels really wrong, ask your parents about seeing a doctor and getting some blood work done.
