  We report photometric estimates of effective temperature, $T_{\rm eff}$,
metallicity, [Fe/H], carbonicity, [C/Fe], and absolute carbon abundances,
$A{\rm (C)}$, for over 700,000 stars from the Southern Photometric Local
Universe Survey (S-PLUS) Data Release 2, covering a substantial fraction of the
equatorial Sloan Digital Sky Survey Stripe 82. We present an analysis for two
stellar populations: 1) halo main-sequence turnoff stars and 2) K-dwarf stars
of mass $0.58 < M/M_{\odot} <0.75$ in the Solar Neighborhood. Application of
the Stellar Photometric Index Network Explorer (SPHINX) to the mixed-bandwidth
(narrow- plus wide-band) filter photometry from S-PLUS produces robust
estimates of the metallicities and carbon abundances in stellar atmospheres
over a wide range of temperature, $4250 < T_{\rm eff} \textrm{(K)} < 7000$. The
use of multiple narrow-band S-PLUS filters enables SPHINX to achieve
substantially lower levels of "catastrophic failures" (large offsets in
metallicity estimates relative to spectroscopic determinations) than previous
efforts using a single metallicity-sensitive narrow-band filter. We constrain
the exponential slope of the Milky Way's K-dwarf halo metallicity distribution
function (MDF), $\lambda_{10, \textrm{[Fe/H]}} = 0.85 \pm 0.21$, over the
metallicity range $-2.5 < \textrm{[Fe/H]} < -1.0$; the MDF of our local-volume
K-dwarf sample is well-represented by a gamma distribution with parameters
$\alpha=2.8$ and $\beta=4.2$. S-PLUS photometry obtains absolute carbon
abundances with a precision of $\sim 0.35$dex for stars with $T_{\rm eff} <
6500$K. We identify 364 candidate carbon-enhanced metal-poor stars, obtain
assignments of these stars into the Yoon-Beers morphological groups in the
$A$(C)-[Fe/H] space, and derive the CEMP frequencies.
