  We report on the results obtained with the global CUPID-0 background model,
which combines the data collected in the two measurement campaigns for a total
exposure of 8.82~kg$\times$yr of $^{82}$Se. We identify with improved precision
the background sources within the 3 MeV energy region, where neutrinoless
double $\beta$-decay of $^{82}$Se and $^{100}$Mo is expected, making more solid
the foundations for the background budget of the next-generation CUPID
experiment. Relying on the excellent data reconstruction, we measure the
two-neutrino double $\beta$-decay half-life of $^{82}$Se with unprecedented
accuracy: $T_{1/2}^{2\nu} = [8.69 \pm 0.05 \textrm{(stat.)}~^{+0.09}_{-0.06}
\textrm{(syst.)}] \times 10^{19}~\textrm{yr}$.
