Interactions between magma oceans and overlying atmospheres on young rocky planets leads to an evolving feedback of outgassing, greenhouse forcing, and mantle melt fraction. Previous studies have predominantly focused on the solidification of oxidized Earth-similar planets, but the diversity in mean density and irradiation observed in the low-mass exoplanet census motivate exploration of strongly varying geochemical scenarios. We aim to explore how variable redox properties alter the duration of magma ocean solidification, the equilibrium thermodynamic state, melt fraction of the mantle, and atmospheric composition. We develop a 1D coupled interior-atmosphere model that can simulate the time-evolution of lava planets. This is applied across a grid of fixed redox states, orbital separations, hydrogen endowments, and C/H ratios around a Sun-like star. The composition of these atmospheres is highly variable before and during solidification. The evolutionary path of an Earth-like planet at 1 AU ranges between permanent magma ocean states and solidification within 1 Myr. Recently solidified planets typically host