I don't quite understand your question about surfaces as the notions of hyperbolicity you are talking about deal with two different structures: a Riemann surface is endowed with a conformal structure, whereas Gromov's hyperbolicity is a property of metric spaces (in particular, of Riemannian manifolds).
There is also some confusion in the torus example: when saying that it is of parabolic type you are talking about its universal cover, whereas when claiming that it is Gromov hyperbolic you are talking about the torus itself (actually it is not really fair to say that a compact metric space is Gromov hyperbolic).
Finally, concerning graphs the answer is no, because you can always attach to your favorite transient planar graph a sequence of circles with increasing radii - it won't change transience and planarity, but will prevent the resulting graph from being hyperbolic.
EDIT: I would still strongly advise against basing any examples or counterexamples on "compact hyperbolic spaces". Although formally they do have the $\delta$-hyperbolicity property, the whole point of developing this theory was to look at the large scale geometry of such spaces, and in particular at their behavior at infinity. If you wish, the notion of a compact hyperbolic space is as rich as that of a compact vector space. This is what I meant by saying that "it is not really fair to say that a compact metric space is Gromov hyperbolic".
As for your revised question, one can formulate it in a more general way: when is a quotient of a Gromov hyperbolic space also Gromov hyperbolic? In order to see its scope, you may first look at the discrete case, where any regular graph is a quotient of the corresponding homogeneous tree.