Tyger:
Let's start with the first question of why a spinor is SU(2). I think you are on the right track, but there are some subtle differences. First for the group theory definitions, as an example
SU(2) -> 2x2 matrix that is unitary with determinant = 1
S0(3) -> 3x3 matrix that is orthogonal with determinant = 1
Non formally, the SU(n) group is 'imaginary' (unitary) matrices, while SO(n) is just the 'real' (orthogonal) matrices.
So a Spinor is a 2 component column vector, this is because for the spin 1/2 particles (proton,neuton,electron,quarks,leptons) they have two states, (spin up) and (spin down). There is more to this due to quantum mechanics and probabilities and so we deal with Pauli spin matrices.
Anyway, to get to the point a spinor is represented by the SU(2) group because
a. n=2, the two components of the 2D vector, spin up, spin down
b. This is the key, the spinor (2D vector) transforms according to the SU(n) group under a coordinate rotation. Note that the 'coordinates' for spinors are related to the probability amplitudes of a measurement. Without going into the math in the forum (maybe I will have time to put an article up on Spinors later) the rotation matrix for spinors is imaginary and is something like exp(-i theta sigma/2) compare this to what you may be use to with normal rotation matrices in mechanics that have sines and cosines these rotation matrices are 'real' and are represented by the group S0(3).
This is a lot of handwaving and you must really work out the math to build understanding. This would make for some great PlanetPhysics articles :)
Finally, the SU(2) is almost identical to the SO(3) group, the difference arising from the need to tie in with physical meaning of QM and the probability amplitude (square of the wave function) so it is not quite the traditional rotation matrix that we are use to in classical mechanics
Question #2: This is what particle physics is all about counting beans :) So we saw above that spinors are a 2D vector and have 1/2 spin and they are classified as Fermions (1/2 integer spin, spin = 1/2 or spin = 3/2). Another type of fermion with spin 3/2 (see Baryon decuplet) will be represented by a different 'rotation' group. The other classification is Bosons (integer spin, spin = 0 or spin = 1).
Bosons
Spin 0 -> Pseudo-scalar mesons
Spin 1 -> Mediators, Vector Mesons
Fermions
Spin 1/2 -> leptons,Quarks, Baryon octet
Spin 3/2 -> Baryon decuplet
To really understand what all this means you need to study QM and particle physics, but I hope this helped a little bit. Please ask any other questions to clear things up and maybe we can crank out a few articles to help clarify things.
Note: I'm sure there might be some mistakes so try to find a good QM or particle physics textbook to use as reference, it has been several years since my particle physics class :) Chow.
Ben
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