I'm a bachelor math student...

>he's an academic

kek

Sets are one of the most natural objects of all mathematics, and there is not much magic to them. Mathematics is founded on sets, basically everything is a set, so you need to be able to manipulate them pretty intuitively. I think the big difference is that we don't tend to see shapes in the plane as "sets", they are just geometric forms. A circle, a ball, a square, whatever. Maybe when you think of a set you think of {1,2,3}. Discrete, finite sets. But shapes and geometric forms are also sets, hell even functions are sets!

you don't have to though. If you want to study computers you don't really need a lot of pure maths, specially pure analysis. Algebra and combinatorics serve you better.

I think I get that, like even a square is just a set of points on the 2d plane, that is continuous between the discrete boundary points

Functions are contiunous, not sets. Also, what you mean by "the discrete boundary points"? The boundary is not discrete, if that's what you mean. A set is discret if you can isolate every element. That is, for each p in A there is an open ball in A that contains only p.

But does the set of all things contain itself?

the "set of all things" is not a set

is there any interesting notion of "higher" quivers, where instead of just vertices and edges, you also have faces, similar to CW complexes?

also how are quiver representations any different from diagrams in the category of R-modules?

The key to succeeding in mathematics professionally is to understand the prerequisite material as second nature. Otherwise, you'll end up reinventing the wheel, which is not what you want for novelty, but is perfectly fine as a hobby. Other than that, it's really a matter of how much time you put into it. Oftentimes, seeing a solution is much easier than getting to it. On the way you there you might find out it's not what you were looking for. At this point, how much of this difference in speed in ruling out or confirming solutions can be attributed to intelligence vs random chance is unclear. Is a more intelligent person faster, better at picking good routes, better able to construct a larger picture, less distracted, more novel, had better instruction, etc.? Without dedication to the field, intelligence really doesn't matter. I hope this offers some clarity in the manner.
Also, don't focus on your colleagues' progress. Some people are great at learning, but that often doesn't translate into being a good researcher. Many a "star" has ended up being useless deadwood without a pipeline. If you do become a star, don't let your advisors coddle you. Helping your colleagues is a great way to test your understanding of the material and is very important as a researcher as your job is to not only discover novel mathematics, but to convince others in your field that you have done so.
>I don't "get" the theory behind math
If you don't understand something, you certainly won't understand that something if you don't try to understand that something. It may be daunting, but you have to start somewhere to get somewhere else.

I don't know of any, sorry, but it seems interesting. I think it might be in line with 2-categories and so on. In fact you can see categories as a kind of quiver, so I guess 2-categories are the thing you describe.

Well in a quiver representation at each vertex you have a vector space, and the whole thing, including it arrows, is a module over the path algebra. I think in a diagram each point is an R-module instead of a vector space.