The use of Type Ia Supernovae (SNe Ia) as cosmological tools has motivated significant effort to: understand what drives the intrinsic scatter of SN Ia distance modulus residuals after standardization, characterize the distribution of SN Ia colors, and explain why properties of the host galaxies of the SNe correlate with SN Ia distance modulus residuals. We use a compiled sample of $sim1450$ spectroscopically confirmed, photometric light-curves of SN Ia and propose a solution to these three problems simultaneously that also explains an empirical 11$sigma$ detection of the dependence of Hubble residual scatter on SN Ia color. We introduce a physical model of color where intrinsic SN Ia colors with a relatively weak correlation with luminosity are combined with extrinsic dust-like colors ($E(B-V)$) with a wide range of extinction parameter values ($R_V$). This model captures the observed trends of Hubble residual scatter and indicates that the dominant component of SN Ia intrinsic scatter is from variation in $R_V$. We also find that the recovered $E(B-V)$ and $R_V$ distributions differ based on global host-galaxy stellar mass and this explains the observed correlation ($gamma$) between mass and Hubble residuals seen in past analyses as well as an observed 4.5$sigma$ dependence of $gamma$ on SN Ia color. This finding removes any need to prescribe different intrinsic luminosities to different progenitor systems. Finally we measure biases in the equation-of-state of dark energy ($w$) up to $|Delta w|=0.04$ by replacing previous models of SN color with our dust-based model; this bias is larger than any systematic uncertainty in previous SN Ia cosmological analyses.
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