  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{\deg} of the line of sight. Moreover, the average Lorentz factor,
<$\Gamma$>= 8-12, is lower than previously suggested for these powerful
sources.
