  In this concluding part of the series of three papers dedicated to the
Swift/BAT hard X-ray survey (BXS), we focus on the X-ray spectral analysis and
statistical properties of the source sample. Using a dedicated method to
extract time-averaged spectra of BAT sources we show that Galactic sources
have, generally, softer spectra than extragalactic objects and that Seyfert 2
galaxies are harder than Seyfert 1s. The averaged spectrum of all Seyfert
galaxies is consistent with a power-law with photon index of 2.00 (+/-0.07).
The cumulative flux-number relation for the extragalactic sources in the 14-170
keV band is best described by a power-law with a slope alpha=1.55 (+/-0.20) and
a normalization of 9.6$\pm1.9 \times 10^{-3}$ AGN deg$^{-2}$ (or 396(+/-80) AGN
all-sky) above a flux level of 2$\times 10^{-11}$erg cm$^{-2}$ s$^{-1}$ (~0.85
mCrab). The integration of the cumulative flux per unit area indicates that BAT
resolves 1-2% of the X-ray background emission in the 14-170 keV band. A
sub-sample of 24 extragalactic sources above the 4.5 sigma detection limit is
used to study the statistical properties of AGN. This sample comprises local
Seyfert galaxies (z=0.026, median value) and ~10% blazars. We find that 55% of
the Seyfert galaxies are absorbed by column densities of Log(N_H)>22, but that
none is a bona fide Compton-thick. This study shows the capabilities of BAT to
probe the hard X-ray sky to the mCrab level.
