Write for simplicity $X=\mathbb{P}^n$. 

An easy way of showing 1) is by noting that $\mathcal I / \mathcal I^2$ injects as a sub bundle of $\mathcal O_Y(-1)^{n+1}$ (this follows from combining the conormal sequence with the Euler sequence) and so none of it's symmetric powers $S^r(\mathcal I / \mathcal I^2)=\mathcal I^r / \mathcal I^{r+1}$ can have any global sections. Now taking the cohomology sequence of
$$
0 \to \mathcal{I}^r/\mathcal{I}^{r+1}\to\mathcal{O}_X/\mathcal{I}^{r+1} \to\mathcal{O}_X/\mathcal{I}^{r}  \to 0,
$$shows that $\Gamma(Y,\mathcal{O}_X/\mathcal{I}^{r+1})$ injects into $\Gamma(Y,\mathcal{O}_X/\mathcal{I}^{r})$, so by induction on $r$, we get 2).