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We derive the most up-to-date Swift-Burst Alert Telescope (BAT) blazar luminosity function in the 14-195 keV range, making use of a clean sample of 118 blazars detected in the BAT 105-month survey catalog, with newly obtained redshifts from the BAT AGN Spectroscopic Survey (BASS). We determine the best-fit X-ray luminosity function for the whole blazar population, as well as for Flat Spectrum Radio Quasars (FSRQs) alone. The main results are: (1) at any redshift, BAT detects the most luminous blazars, above any possible break in their luminosity distribution, which means we cannot differentiate between density and luminosity evolution; (2) the whole blazar population, dominated by FSRQs, evolves positively up to redshift z~4.3, confirming earlier results and implying lower number densities of blazars at higher redshifts than previously estimated. The contribution of this source class to the Cosmic X-ray Background at 14-195 keV can range from 5-18%, while possibly accounting for 100% of the MeV background. We also derived the average 14 keV-10 GeV SED for BAT blazars, which allows us to predict the number counts of sources in the MeV range, as well as the expected number of high-energy (>100 TeV) neutrinos. A mission like COSI, will detect 40 MeV blazars and 2 coincident neutrinos. Finally, taking into account beaming selection effects, the distribution and properties of the parent population of these extragalactic jets are derived. We find that the distribution of viewing angles is quite narrow, with most sources aligned within < 5° of the line of sight. Moreover, the average Lorentz factor, <$Γ$>= 8-12, is lower than previously suggested for these powerful sources.