  Reionization is thought to be driven by faint star-forming galaxies, but
characterizing this population has long remained very challenging. Here we
utilize deep nine-band NIRCam imaging from JADES to study the star-forming and
ionizing properties of 756 $z\sim6-9$ galaxies, including hundreds of very
UV-faint objects ($M_\mathrm{UV}>-18$). The faintest ($m\sim30$) galaxies in
our sample typically have stellar masses of $M_\ast\sim(1-3)\times10^7$
$M_\odot$ and young light-weighted ages ($\sim$50 Myr), though some show strong
Balmer breaks implying much older ages ($\sim$500 Myr). We find no evidence for
extremely massive galaxies ($>3\times10^{10}$ $M_\odot$). We infer a strong
(factor $>$2) decline in the typical [OIII]$+$H$\beta$ EWs towards very faint
$z\sim6-9$ galaxies, yet a weak UV luminosity dependence on the H$\alpha$ EWs
at $z\sim6$. We demonstrate that these EW trends can be explained if fainter
galaxies have systematically lower metallicities as well as more
recently-declining star formation histories relative to the most UV-luminous
galaxies in our sample. Our data provide evidence that the brightest galaxies
are frequently experiencing a recent strong upturn in SFR. We also discuss how
the EW trends may be influenced by a strong correlation between $M_\mathrm{UV}$
and Lyman continuum escape fraction. This alternative explanation has
dramatically different implications for the contribution of galaxies along the
luminosity function to cosmic reionization. Finally, we quantify the
photometric overdensities around two $z>7$ strong Ly$\alpha$ emitters. One
Ly$\alpha$ emitter lies close to a strong photometric overdensity while the
other shows no significant nearby overdensity, perhaps implying that not all
strong $z>7$ Ly$\alpha$ emitters reside in large ionized bubbles.
