We present exoplanet occurrence rates estimated with approximate Bayesian computation for planets with radii between 0.5 and 16 $R_{bigoplus}$ and orbital periods between 0.78 and 400 days, orbiting FGK dwarf stars. We base our results on an independent planet catalogue compiled from our search of all ~200,000 stars observed over the Kepler mission, with precise planetary radii supplemented by Gaia DR2-incorporated stellar radii. We take into account detection and vetting efficiency, planet radius uncertainty, and reliability against transit-like noise signals in the data. By analyzing our FGK occurrence rates as well as those computed after separating F-, G-, and K-type stars, we explore dependencies on stellar effective temperature, planet radius, and orbital period. We reveal new characteristics of the photoevaporation-driven radius gap between ~1.5 and 2 $R_{bigoplus}$, indicating that the bimodal distribution previously revealed for $P$ < 100 days exists only over a much narrower range of orbital periods, above which sub-Neptunes dominate and below which super-Earths dominate. Finally, we provide several estimates of the eta-Earth value -- the frequency of potentially habitable, rocky planets orbiting Sun-like stars. For planets with sizes 0.75 - 1.5 $R_{bigoplus}$ orbiting in a conservatively defined habitable zone (0.99 - 1.70 AU) around G-type stars, we place an upper limit (84.1th percentile) of <0.18 planets per star.
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