Im not sure if this counts as a full answer, but it is a nice example which will hopefully shed light on some of your questions.
The canonical divisor $K$ of an Enriques surface satisfies $2K=0$, but $K\neq 0$ in the Picard group. It follows that $K$ is not the restriction of any divisor in $\mathbb{P}^n$, as these are necessarily not torsion.