Quantum Mechanics - Pauli Spin Matrices

The Pauli Spin matrices:

$$\sigma_1=\left[ \begin{array}{ c c } 0 & 1 \\ 1 & 0 \end{array} \right],\sigma_2=\left[ \begin{array}{ c c } 0 & -i \\ i & 0 \end{array} \right],\sigma_3=\left[ \begin{array}{ c c } 1 & 0 \\ 0 & -1 \end{array} \right]$$

are used to represent the angular momentum operators for spin 1/2 in 3 dimensions for a single particle. My question is how are they derived? Particularly, what properties of the matrices correspond to general features of angular momentum and which are specific to spin half?

If we had two spin-half particles (A and B say) with basis states $$|\uparrow>_{A}|\uparrow>_{B},|\downarrow>_{A}|\uparrow>_{B},|\uparrow>_{A}|\downarrow>_{B},|\downarrow>_{A}|\downarrow>_{B}$$ then is it possible to find matrix representations of the spin matrices for each of A and B in the three axes? I'm guessing they would be 4x4 matrices since we have four basis states.

I know that's a lot to ask but the wikipedia page on Pauli matrices hardly mentions their relevance to quantum mechanics until the end of the page.