In the third APOKASC catalog, we present data for the complete sample of 15,808 evolved stars with APOGEE spectroscopic parameters and Kepler asteroseismology. We used 10 independent asteroseismic analysis techniques and anchor our system on fundamental radii derived from Gaia L and spectroscopic Teff. We provide evolutionary state, asteroseismic surface gravity, mass, radius, age, and the data used to derive them for 12,418 stars. This includes 10,036 exceptionally precise measurements, with median fractional uncertainties in {nu}_max_, {Delta}{nu}, mass, radius, and age of 0.6%, 0.6%, 3.8%, 1.8%, and 11.1%, respectively. We provide more limited data for 1624 additional stars that either have lower-quality data or are outside of our primary calibration domain. Using lower red giant branch (RGB) stars, we find a median age for the chemical thick disk of 9.14+/-0.05(ran) +/-0.9(sys) Gyr with an age dispersion of 1.1Gyr, consistent with our error model. We calibrate our red clump (RC) mass loss to derive an age consistent with the lower RGB and provide asymptotic GB and RGB ages for luminous stars. We also find a sharp upper-age boundary in the chemical thin disk. We find that scaling relations are precise and accurate on the lower RGB and RC, but they become more model dependent for more luminous giants and break down at the tip of the RGB. We recommend the use of multiple methods, calibration to a fundamental scale, and the use of stellar models to interpret frequency spacings.