  We obtain orbital counting results for the class of strongly hyperbolic
metrics on hyperbolic groups. To achieve this we combine ergodic theoretic
techniques involving the Mineyev topological flow and symbolic dynamics. Our
results apply to the Green metric associated to an admissible, finitely
supported, symmetric random walk and to the Mineyev hat metric. We also
describe the domain of analyticity for the Poincar\'e series associated to
these metrics, prove mixing results for the Mineyev topological flow and obtain
correlation asymptotics for pairs of metrics.
