I think both of your questions can by answered positive. Since the Hopf fibration is an $S^1$ bundle it comes as a sphere bundle of complex line bundle over $S^2$. Since the map $S^3 \to S^2$ is the generator of $\pi_3(S^2)$ the first Chern class of that line bundle must be a generator of $H^2(S^2;\mathbb Z)$ which determines the isomorphism type of the line bundle.

Moreover a circle action on $S^2$ can be lifted to a line bundle if and only if the first Chern class possess an equivariant extension in $H_{S^1}(S^2;\mathbb Z)$ which here is always true.

I think both of your questions can by answered positive. Since the Hopf fibration is an $S^1$ bundle it comes as a sphere bundle of complex line bundle over $S^2$. From the Gysin sequence the first Chern class of that line bundle must be a generator of $H^2(S^2;\mathbb Z)$ which determines the isomorphism type of the line bundle.

Moreover a circle action on $S^2$ can be lifted to a line bundle if and only if the first Chern class possess an equivariant extension in $H_{S^1}(S^2;\mathbb Z)$ which here is always true.

Edit: I would like to remark that lifting actions from the base space to a vector bundle is in general a very difficult problem (which is solved for complex line bundles) but unknown (as far as I know) for complex vector bundle of rank higher than 2. There are some results for oriented vector bundles over spheres.