The recurrent nova M31N 2008-12a experiences annual eruptions, contains a near-Chandrasekhar mass white dwarf, and has the largest mass accretion rate in any nova system. In this paper, we present Hubble Space Telescope (HST) WFC3/UVIS photometry of the late decline of the 2015 eruption. We couple these new data with archival HST observations of the quiescent system and Keck spectroscopy of the 2014 eruption. The late-time photometry reveals a rapid decline to a minimum luminosity state, before a possible recovery / re-brightening in the run-up to the next eruption. Comparison with accretion disk models supports the survival of the accretion disk during the eruptions, and uncovers a quiescent disk mass accretion rate of the order of $10^{-6}\,M_\odot\,\mathrm{yr}^{-1}$, which may rise beyond $10^{-5}\,M_\odot\,\mathrm{yr}^{-1}$ during the super-soft source phase - both of which could be problematic for a number of well-established nova eruption models. Such large accretion rates, close to the Eddington limit, might be expected to be accompanied by additional mass loss from the disk through a wind and even collimated outflows. The archival HST observations, combined with the disk modeling, provide the first constraints on the mass donor; $L_\mathrm{donor}=103^{+12}_{-11}\,L_\odot$, $R_\mathrm{donor}=14.14^{+0.46}_{-0.47}\,R_\odot$, and $T_\mathrm{eff, donor}=4890\pm110$ K, which may be consistent with an irradiated M31 red-clump star. Such a donor would require a system orbital period $\gtrsim5$ days. Our updated analysis predicts that the M31N 2008-12a WD could reach the Chandrasekhar mass in