A good reference for 1) is Bourbaki: Lie groups and Lie algebras Chapter 9. See in particular Section 4.6. 

In particular it follows that 2) $\pi_1(G/T) = 0$ and that 4) $\pi_1(G)$ is finite if and only if $G$ is semisimple.  

Concerning 3) $\pi_2(G) = 0$ always, which is a theorem of Cartan. I don't recall Cartan's proof, but it follows from Bott's analysis of the cell structure of G/T, and can also be proved using that $H^*(G)$ is a Hopf algebra (See Browder: Torsion in H-spaces. Ann. of Math. (2) 74 1961 24–51.).

EDIT (3 years later..): Just to elaborate, 1) is completely answered the above Bourbaki reference. The formula is that $\pi_1(G) = L/L_0$, where L is the integral lattice from above and $L_0$ is the coroot lattice. For the connoisseurs out there I mention that there is also a homotopical version, in that the formula also holds for p-compact groups (see Section 8 of my paper with Kasper Andersen on the classification of 2-compact groups linked here: http://www.math.ku.dk/~jg/papers/2classification.pdf)