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.).