The third Gaia data release (DR3) contains high-precision, sparse-in-time brightness measurements of over 150000 asteroids. We employed a light-scattering inversion technique to estimate the rotation periods, spin pole orientations, shapes, and photometric phase function parameters (slope and absolute magnitude) of over 8000 asteroids based solely on DR3 photometry. Using triaxial ellipsoid and convex shapes, we sought the best-fit shape, spin, and linear slope, along with their uncertainties, via a Markov chain Monte Carlo sampling technique. We also fit HG12 and HG1G2 phase functions from predicted brightnesses derived from the fit shapes. Using previously reported diameters, the Gaia G-band geometric albedos were calculated. Variations in the spin and shape properties were assessed among various families and background populations of the Main Belt. We found that for the vast majority of our objects the best-fit ellipsoid spin poles are comparable to that of a convex shape. We rejected 15% of convex shapes and used the acceptable solutions to investigate differences in the shape distribution of families in the Main Belt. Revisiting the amplitude phase relationship, we found a strong dependence on the shape b/a elongation. The Bond albedo was calculated from the phase integral and shown to correlate well with the G12 slope parameter and known taxonomic classifications. The G-band absolute magnitudes and geometric albedos are systematically fainter than V-band values. The assumption of ellipsoid shape for sparse datasets is sufficient for estimating the spin pole longitude and latitude. We find no correlation between the shapes and spins of main-belt asteroid families and their ages. Absolute magnitudes and phase functions derived from Gaia photometry should be favored over the V band when estimating the solar energy budget, such as in thermal modeling applications. Asteroid taxonomies can be assessed to some degree from the photometric slopes and albedos.