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