Determining which small exoplanets have stony-iron compositions is necessary for quantifying the occurrence of such planets and for understanding the physics of planet formation. Kepler-10 hosts the stony-iron world Kepler-10b (K10b), and also contains what has been reported to be the largest solid silicate-ice planet, Kepler-10c (K10c). Using 220 radial velocities (RVs), including 72 precise RVs from Keck-HIRES of which 20 are new from 2014-2015, and 17 quarters of Kepler photometry, we obtain the most complete picture of the Kepler-10 system to date. We find that K10b (Rp=1.47 Re) has mass 3.72$pm$0.42 Me and density 6.46$pm$0.73 g/cc. Modeling the interior of K10b as an iron core overlaid with a silicate mantle, we find that the iron core constitutes 0.17$pm$0.11 of the planet mass. For K10c (Rp=2.35 Re) we measure Mp=13.98$pm$1.79 Me and $rho$=5.94$pm$0.76 g/cc, significantly lower than the mass computed in Dumusque et al. (2014, 17.2$pm$1.9 Me). Internal compositional modeling reveals that at least $10%$ of the radius of Kepler-10c is a volatile envelope composed of hydrogen-helium ($0.2%$ of the mass, $16%$ of the radius) or super-ionic water ($28%$ of the mass, $29%$ of the radius). Analysis of only HIRES data yields a higher mass for K10b and a lower mass for K10c than does analysis of the HARPS-N data alone, with the mass estimates for K10c formally inconsistent by 3$sigma$. Splitting the RVs from each instrument leads to inconsistent measurements for the mass of planet c in each data set. This suggests that time-correlated noise is present and that the uncertainties in the planet masses (especially K10c) exceed our formal estimates. Transit timing variations (TTVs) of K10c indicate the likely presence of a third planet in the system, KOI-72.X. The TTVs and RVs are consistent with KOI-72.X having an orbital period of 24, 71, or 101 days, and a mass from 1-7 Me.
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