We present accurate fully quantum calculations of thermal rate constants for a symmetric double well system coupled to a dissipative bath. The calculations are performed using the quasiadiabatic propagator path integral (QUAPI) methodology to evaluate the flux-flux correlation function whose time integral determines the rate coefficient. The discretized path integral converges very rapidly in the QUAPI representation, allowing efficient calculation of quantum correlation functions for sufficiently long times. No ad hoc assumption is introduced and thus these calculations yield the true quantum mechanical rate constants. The results presented in the paper demonstrate the applicability of the QUAPI methodology to practically all regimes of chemical interest, from thermal activation to deep tunneling, and the quantum transmission factor exhibits a Kramers turnover. Our calculations reveal an unusual step structure of the integrated reactive flux in the weak friction regime as well as quantum dynamical enhancement of the rate above the quantum transition state theory value at low temperatures, which is largely due to vibrational coherence effects. The quantum rates are compared to those obtained from classical trajectory simulations. We also use the numerically exact classical and quantum results to establish the degree of accuracy of several analytic and numerical approximations, including classical and quantum Grote-Hynes theories, semiclassical transition state theory (periodic orbit) estimates, classical and quantum turnover theories, and the centroid density approximation.