Let $G$ be a compact connected Lie group and let $T$ be a maximal torus of $G$, with Lie algebras $\frak{g}$ and $\frak{t}$ respectively.  Then, $\frak{t}$ can be considered as a Cartan subalgebra of $\frak{g}$.  It is known that  the kernel of exponential map $exp : \frak{t} \to$ $T$ **is the lattice of all integral weights of $\frak{g}$**, i.e. weihts $\lambda \in (it)^*$ such that $\lambda(H)\in 2\pi i\mathbb{Z},$  whenever $exp H= I$ for $H\in\frak{t}$.

I have the following questions: 

1) What is the **relation** between  **the first fundamental group** $\pi_{1}(G)$ of $G$ with the **integral lattice** described above? I am trying to find any good references about this fact, but it seems difficult. 

2) How we can use the fibration $T\to G$$\to G/T$    to compute  $\pi_{1}(G/T)$?

3) What we can say about  **the second fundamental group** $\pi_{2}(G)$?

4) Is it true, that if $G$   is **semisimple**, then $\pi_{1}(G)$ is finite? 
Thank you!