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.