I would say those questions are for recent graduates who want a junior quant role.
What I mean is that you should be learning basics maths anyway, and when you know a bit you can solve all those questions.
What a lot of people could benefit from is learning how to learn.
EDIT: Thanks for downvotes guys. Should I also downvote when someone says "don't be discouraged with distributed systems, or asynchronous programming" etc? I don't know those things, so maybe I should downvote those people?
Anyway, thanks for the feedback.
Those fluent in math really need to drop this egocentric view. If you can't do matrix manipulations you aren't curious about the world? Give me a break. The breadth of topics that yield understanding of various parts of our world is vast. Do you have an early university understanding of history, politics, math, sociology, psychology, economics, law, capitalism, etc. etc. etc.
It's pretty high up there in "list of things to learn if you're curious about the world". Of course, so are history, economics, etc'
Parts 2 and 3 are probability and statistics, but they are the kind of advanced statistics that is typically only taught to math students. Second and third year university courses in probability and statistics would let you answer these questions.
Part 4 is statistical mechanics, I think. That's a third-year physics course, and one of the more mathematically demanding ones, at that.
Part 5 is specific to finance. The course you are looking for is probably called Mathematical Finance.
The conventional route to learning all this stuff is an undergraduate degree in mathematics, with an emphasis on statistics. Now, getting there is a long road, but you have to start somewhere. You might possibly be able to learn it all on your own, but it's easy to get stuck and hard to get unstuck without a teacher to help you.
If you are in the US, start by taking a couple of classes in a community college. In your case, the course you are looking for is probably called Pre-Calculus or maybe Algebra II. If things go well you can continue from there.
As for what use this is, apparently you can get a good job as a quantitative financial analyst if you know this stuff.