I tried to read the first page and I failed miserably. You all sound fluent in math!
Also, @Holocaust, would you happen to have a link to that blog? I’m kind of interested in reading it.
@Halogen reply to your first post:
you need a good amount of analysis to prove the fundamental theorem of calculus… heck, you need at least: basic set theory, basic group theory, vector spaces, topology, linear algebra, a lot of rigorous mapping theorems, and a lot of definitions and proofs in general just to be able to define integration. it would be insane if you could prove that theorem by yourself without knowledge of any of these fields - that would be replicating a hundred years of mathematical work!
a ring is something that if you find that you are too attached to, it’s probably bad and you should chuck it in mount doom before it consumes you. it can also mean an algebraic structure which basically generalizes some of the properties of addition and multiplication into really abstract notation. you should make sure you can distinguish them from like, monoids, groups, fields, ideals, algebras, etc. - a lot of subtle differences that make a huge impact!
what is the twelvefold way? oops
and yes logic is hard 
but do note that all the math we have now is a result of hundreds of years of efforts by mathematicians all around the world. it would be impossible to try to learn everything that has been found - heck, it’s hard enough just to get to the 1900s. math is one of those fields that is already extensively developed, so it will feel like you’re lost and can’t do anything for a long time (unless you’re like terence tao)
as for what you should do with your life, the next episode of the anime i’m starring in is coming out in around 14 hours. you should watch it.
EDIT: i assumed you were in high school. i would like to change the tone of my post but i don’t know how to edit posts. oops. but the best i can say is that (1) math is too hard and (2) be like fermat and do other stuff while thinking about math if you’re feeling lost? like you can go make tons of money while thinking about math problems in the back of your head, so even if math is too hard, you can still do cool stuff.