Question

Fields are one of the following: scalars, vectors, spinors or some Lie algebra elements, right? And it's often said that scalars are spin-0 and vectors are spin-1. So, what's idea of correspondence between nature of field and it's spin value?

Answer 1

A classical field is a section of a vector bundle on the space-time manifold $M$.

That vector bundle is typically obtained by using the associated bundle construction applied to the frame bundle of $M$, and some irreducible representation of SO(d) (you've noted that I'm in Eucledian signature).

Sometimes, this is not enough, and one has to start with a double cover of the frame bundle of $M$, and use some irreducible representation of the double cover Spin(d) of SO(d).

So, the types of field (scalar, vector, ...) are in one-to-one correspondence with types of irreducible representations of the spin group Spin(d).

Those irreducible representations are classified.
For d=3, there is exactly one representation of each dimension. The irreducible representations of the group Spin(3)= SU(2) are:

spin-0 = 1-dimensional trivial irrep
spin-1/2 = 2-dimensional irrep
spin-1 = 3-dimensional irrep
etc.

For d=4, we have an isomorphism between Spin(4) and SU(2)xSU(2), and so irreducible representations are classified by pairs of non-negative half-integers.