Where are the theorems and the proofs? Can the usual theorems of the "year of linear algebra" be proved using these arrows?

To understand what's happening here, I'd recommend you to first understand Representation theory: https://en.wikipedia.org/wiki/Representation_theory

In representation theory we reduce problems of algebra to problems of linear algebra. E.g. the standard example is to find representations of groups, this way we can represent group operations as matrix operations. We do this because (1) linear algebra is mathematically very well-understood, (2) in terms of applications, linear algebra is computationally fast, faster than implementing the group with code manually (at least, in general).

In the OP post, author reduces quiver (which is a particular kind of algebra) to linear algebra. Once this is done, the intention is to solve problems of quivers in the language of linear algebra.