We present new optical and near-infrared images of the hydrogen-poor planetary nebula Abell 30 and produce detailed models that account for the major observed morphological and IR properties. By imaging the nebula in the K band, we confirm the presence of hot dust in an expanding equatorial ring of H-poor gas. No emission was detected from the H-poor polar knots, suggesting a dust deficiency htere relative to the equatorial ring. The near-IR emission is attributed to small carbonaceous dust grains which are stochastically heated by stellar ultraviolet photons. Using an adaptive version of a matrix method devised by Guhathakurta & Draine (1989) to model stochastic heating, we find that the near-IR spectrum observed by Dinerstein & Lester (1984) requires the presence of dust grains down to approximately 0.0007 microns in radius. This minimum grain radius is in excellent agreement with our calculations of the grain destruction by energetic stellar UV photons: we find that carbon clusters with less than approximately 140 atoms (0.0007 microns in radius) are destroyed by stellar UV photons in approximately 1000 yr, the kinematic age of H-poor ejecta. Modeling of the far-IR dust emission implies that the bulk of the dust mass in A30 must reside at distances several times greater than the distance of the equatorial ring from the central star. This spatial dust distribution is attributed to the interaction of the stellar wind with the inhomogeneous H-poor ejecta. Most of the H-poor gas and dust has been apparently carried outward by the stellar wind, leaving behing dense, H-poor knots with prominent wind-blown tails in the equatorial ring and in the polar knots. This picture is supported by the presence of a stellar wind-blown bubble within the H-rich envelope in our optical images.