How is spin defined in Quantum Mechanics, exactly? I know that spin gets a proper and complete definition in Quantum Field Theory, when we account for relativity in our quantum theory. This question is not about this.
I am instead interested in understanding how spin is defined in non relativistic quantum mechanics, assuming we know nothing about Dirac's equation and such.
Regarding this question I know that in "vanilla" QM spin is somewhat introduced by force, but I don't exactly know what this means: is spin introduced in the theory only on the base of experimental evidence, such as Stern-Gerlach experiment? Or maybe there is some other experimental evidence? Or other reasons to introduce it?
Anyway then, by some reasoning, we determine that it is a good idea to introduce spin, as an intrinsic property of particles, and to postulate that it has the algebric structure of angular momentum. This pratically means that the following is postulated to be true:
$$[S_i,S_j]=i\hbar \varepsilon _{ijk}S_k \tag{1}$$
$$[S^2,S_i]=0 \tag{2}$$
this should be it. No other postulate. But then something strange is stated, and it is stated not as a postulate but somehow as a consequence of what we have said so far:

Particles with spin 1/2 are associated with angular momentum in two dimensions and particle with spin 1 are associated with angular momentum in three dimensions.

Why? What does this mean exactly?
But the strangeness is not over. Then we are somehow able to show that spin $1/2$ particles have spin operators represented by Pauli Matricies:
$$S_i=\frac{\hbar}{2}\sigma _i$$
and we are also able to show that spin $1$ particles have spin operators represented by another set of matricies, these matricies as far as I know don't have a name, but they are 3x3, in accordance with what we have stated in the citation.
Is all this simply postulated? Or it is derived from (1),(2) as I understood? And if it is indeed derived from (1),(2): how exactly is it derived? How can we find out that the spin is represented by these matricies in particular? And also why spin $1/2$ is in 2D and spin $1$ is in 3D?
This bit in particular seems really strange also because in the case of spin $1/2$ spin can be measured either up or down so I can kinda understand why it is in 2D, but what about spin $1$ particles?
 A: 
Particles with spin 1/2 are associated with angular momentum in two dimensions and particle with spin 1 are associated with angular momentum in three dimensions.

This is flat wrong and ambiguous nonsense, at best,  and you should toss the sloppy text you saw it in. It is pig Latin group theory. A good introduction to Lie Group theory might well be  in order. I do understand this is precisely what you aim to avoid, but it is a little bit like asking to bypass calculus and still utilize its techniques. The best you could ask for is a gentle introduction.
Your confusions arise from the newbie efforts of 1920s physicists to understand quantum angular momentum and spin, and how they enter the Lorentz group and their theories.
Both spin 1 and spin 1/2 (and higher integral or half-integer spins for that matter) are associated with the very same Lie  algebra structure (1) you wrote down. ((2) is an easy consequence thereof.) The three ("generators") operators $S_i$ of this algebra, when suitably exponentiated describe the group of rotations and an associated Lie group with the same Lie algebra.
It turns out that, quite independently of physics, by ineluctable mathematical necessity, these operators may be irreducibly represented by 2×2, 3×3, 4×4, 5×5,... matrices, acting on spaces of 2d, 3d, 4d, 5d, ... vectors. The dimensionality of these vector spaces correspond to spin s = 1/2, 1, 3/2,2, etc. (D=2s+1). The operator $S^2$ in your (2) has characteristic different "eigenvalues" for each such irrep, namely numbers multiplying the identity matrix in each  "space" dimension D=2s+1:   $~~~S^2=s(s+1) 1\!\!1$. Do study these matrices, which certainly include the spin-1 ones you are asking about.
These spaces may represent peculiar internal symmetries such as isospin, etc, but, in spacetime, the 3d representation corresponds to our three space dimensions in which we rotate, and the 2d irrep to an abstract complex 2d "spinor space", much unlike our three space dimensions, so talking about it in the same breath as the three space dimensions as your pestiferous quote is bound to confuse you.
This marvelous group theory was invented/discovered out of pure reason in the   19th century, and, when QM emerged in the 20th, physicists had the ready tools to recognize it described the selection rules and Zeeman phenomena involved. Trying to "derive" it from physical "axioms" is as silly as trying to derive matrix calculus, or even geometry, out of physics. Because, at the time, physicists were not too familiar with Lie algebras, sophisticates like Wigner and Dirac (brothers in law) made it easy for them to apply these structures to QM without undue formalism; but, at the end of the day, you'd best start with the elegant and tight mathematical theory, and just apply it to physics, almost magically fitting it--the way geometry does.
A: You are aware that spin arises naturally from Dirac's equations, but classically the justification for spin was historically more ad-hoc. The role that spin plays is well described by anna v.
What is spin? Whenever you ask this sort of question in QM you get into awkward territory, but here goes. It is the property that is measured by operators which obey
$$[L_i, L_j] = i \hbar e_{ijk} L_k$$
It is one of the simplest non-trivial algebras, and has some beauty. For example, it does not rely on any co-ordinate system, and it has a natural scale. I.e. if you change $L_i \rightarrow \lambda.L_i$ is instantly detectable.
You don't need group theory to understand the representation (but it helps). For 1/2 spin, define 2 states
\begin{bmatrix} 1 \\ 0 \end{bmatrix} and \begin{bmatrix} 1 \\ 0 \end{bmatrix}
and look for 3 2x2 matrices that operator on them and obey the commutator relations, If you play around enough, the Pauli matrices pop out. Some algebra shows in this case the angular momentum is $\pm\hbar/2$ (spin 1/2).
Similarly, if you have a particle with 3 possible states, you get different matrices (obviously) and spin takes the values $-\hbar, 0, \hbar$.
The statement "Particles with spin 1/2 are associated with angular momentum in two dimensions and particle with spin 1 are associated with angular momentum in three dimensions" is confused. Spin 1/2 particles are associated with a spin space that is complex and 2D, they do not directly correspond to 2D space (or space-time).
