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 bundle $\omega$ of an Enriques surface satisfies $\omega \otimes \omega=0$, but $\omega\neq 0$ in the Picard group. It follows that $\omega$ is not the restriction of any line bundle in $\mathbb{P}^n$, as these can't be non-zero torsion.