For me, the magnetic vector potential $\vec A$ is the fundamental quantity, from which the magnetic field $\vec B = \vec \nabla \times \vec A$ and the magnetic energy $E = - \vec \mu \cdot \vec B$ is derived. This opinion is due to the Aharonov-Bohm effect. However, I don't think that the question "which physical quantity is more fundamental" is very fruitful, as one quickly enters philosophical discussions and looses the fact, that different physical quantities are useful to describe different aspects of nature. For example, the vector potential $\vec A$ can be expressed as a line integral over the current density $\vec J$. Why should $\vec A$ be more fundamental than $\vec J$?
To your second question, I remember from theoretical mechanics, that "any vector fields can be described by a sum of two potentials: the curl of a potential and the gradient of a potential" (please correct me if I am wrong). Vector fields which can be expressed as gradients of a potential -- such as $\vec
E = -\vec \nabla \phi$, where $\vec E$ is the electric field and $\phi$ is its potential -- are called conservative, as their line integral depends only on the start and then end-point, but not on the chosen path between these points. In contrast the magnetic field $\vec B = \vec \nabla \times \vec A$ is not conservative. The obtained effect depends on the chosen path.
I know, that this does not really answer your questions. However, these comments are simply to long to post them as such.