Line bundle defined by exceptional divisor; $\mathcal O(E) = \pi^*\mathcal O(-1)$

Question

In Huybrechts book (Cor 2.5.6), it is shown:

Let $X=\mathbb C^{n}$ and $\hat X$ its blow-up at $0$. Then locally $\mathcal O(E) = \pi^*\mathcal O(-1)$, where $\pi: \hat X \to \mathbb P^{n-1}$.

I have trouble understanding the proof:

1. Its clear, that $\bigcup_{(\ell,z)\in \hat X} \ell$ is a line bundle. But why is it isomorphic to $\mathcal O(E)$?
2. Why does the section $t(\ell,z)=((\ell,z),z)$ vanish with multiplicity $1$?
3. Why does 2. imply $\mathcal O(E) = \pi^*\mathcal O(-1)$?

I think 2. is simply because the function $z$ vanishes with multiplicity $1$ at $0$, but for the rest I have no clue.