We investigate perfect matchings and essential spanning forests in planar hyperbolic graphs via circle packings. We prove the existence of nonconstant harmonic Dirichlet functions that vanish in a closed set of the boundary, generalizing a result in \cite{bsinv}. We then prove the existence of extremal infinite volume measures for uniform spanning forests with partially wired boundary conditions and partially free boundary conditions, generalizing a result in \cite{BLPS01}. Using the double circle packing for a pair of dual graphs, we relate the inverse of the weighted adjacency matrix to the difference of Green's functions plus an explicit harmonic Dirichlet function. This gives explicit formulas for the probabilities of any cylindrical events. We prove that the infinite-volume Gibbs measure obtained from approximations by finite domains with exactly two convex white corners converging to two distinct points along the boundary is extremal, yet not invariant with respect to a finite-orbit subgroup of the automorphism group. We then show that under this measure, a.s.~there are no infinite contours in the symmetric difference of two i.i.d.~random perfect matchings. As an application, we prove that the variance of the height difference of two i.i.d.~uniformly weighted perfect matchings under the boundary condition above on a transitive nonamenable planar graph is always finite; in contrast to the 2D uniformly weighted dimer model on a transitive amenable planar graph as proved in \cite{RK01,KOS06}, where the variance of height difference grows in the order of $\log n$, with $n$ being the graph distance to the boundary. This also implies that a.s.~each point is surrounded by finitely many cycles in the symmetric difference of two i.i.d.~perfect matchings, again in contrast to the 2D Euclidean case.