This paper investigates the downlink performance of K-tier heteregeneous cellular networks (HCNs) under general settings. First, Gaussian approximation bounds for the standardized aggregate wireless interference (AWI) in dense K-tier HCNs are obtained for when base stations (BSs) in each tier are distributed over the plane according to a spatial and general Poisson point process. The Kolmogorov-Smirnov (KS) distance is used to measure deviations of the distribution of the standardized AWI from the standard normal distribution. An explicit and analytical expression bounding the KS distance between these two distributions is obtained as a function of a broad range of network parameters such as per-tier transmission power levels, per-tier BS intensity, BS locations, general fading statistics, and general bounded path-loss models. Bounds achieve a good statistical match between the standardized AWI distribution and its normal approximation even for moderately dense HCNs. Second, various spatial performance metrics of interest such as outage capacity, ergodic capacity and area spectral efficiency in the downlink of K-tier HCNs for general signal propogation models are investigated by making use of the derived distribution approximation results. Considering two specific BS association policies, it is shown that the derived performance bounds track the actual performance metrics reasonably well for a wide range of BS intensities, with the gap among them becoming negligibly small for denser HCN deployments.