The Zwicky Transient Facility Census of the Local Universe survey yielded a sample of 330 Type IIP supernovae (SNe) with well-constrained peak luminosities. In paper I, we measured their luminosity function and volumetric rate. Here (paper II), we present the largest systematic study of lightcurve properties for Type IIP SNe from a volume-limited survey, analyzing a selected subset of 129 events, including 16 low-luminosity Type IIP (LLIIP) SNe with M_r,peak ≥ −16 mag. We find that plateau slope correlates with peak brightness, with many LLIIP SNe showing positive slopes—suggesting smaller progenitor radii and distinct density profiles compared to brighter Type IIP SNe. The plateau duration shows only a weak dependence on peak brightness, likely suggesting binary interaction. One SN exhibits a plateau-to-tail drop of >3.5 mag, consistent with an electron-capture or failed SN with very low or zero nickel mass. We derive explosion and progenitor parameters of the entire Type IIP SN sample using semi-analytical and radiation-hydrodynamical models. Based on radiation-hydrodynamical model fitting, LLIIP SNe are characterized by low nickel masses (0.001–0.025 M_⊙), low explosion energies (0.1–0.28 × 10⁵¹ erg), low ejecta masses (8.1 ^+0.8 _-1.7 M_⊙), and ZAMS masses below 11 M_⊙. In comparison, the full Type IIP SN sample spans a wider range with nickel masses (0.001–0.222 M_⊙), explosion energies (0.10-4.43 × 10⁵¹ erg), ejecta masses (5.4–24.8 M_⊙), and ZAMS masses (9.3-16.7 M_⊙). We find strong correlations between peak brightness, explosion energy, and nickel mass that extend to the low-luminosity end. We conclude that LLIIP SNe represent the faint, low-energy end of the Type IIP population and originate from the lowest-mass core-collapse progenitors.