We report photometric estimates of effective temperature, T eff, metallicity, [Fe/H], carbonicity, [C/Fe], and absolute carbon abundances, A(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 o˙ < 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 temperatures, 4250 < T eff(K) < 7000. The use of multiple narrow-band S-PLUS filters enables SPHINX to achieve substantially lower levels of "catastrophic failures"(i.e., 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), λ 10,[Fe/H] = 0.85 ± 0.21, over the metallicity range -2.5 < [Fe/H] < -1.0; the MDF of our local-volume K-dwarf sample is well-represented by a gamma distribution with parameters α = 2.8 and β = 4.2. S-PLUS photometry obtains absolute carbon abundances with a precision of ∼0.35 dex for stars with T 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 we derive the CEMP frequencies.