Large dynamic stresses near earthquake rupture fronts may induce an inelastic response of the surrounding materials, leading to increased energy absorption that may affect dynamic rupture. We systematically investigate the effects of off-fault plastic energy dissipation in 2-D in-plane dynamic rupture simulations under velocity-and-state-dependent friction with severe weakening at high slip velocity. We find that plasticity does not alter the nature of the transitions between different rupture styles (decaying versus growing, pulse-like versus crack-like, and subshear versus supershear ruptures) but increases their required background stress and nucleation size. We systematically quantify the effect of amplitude and orientation of background shear stresses on the asymptotic properties of self-similar pulse-like ruptures: peak slip rate, rupture speed, healing front speed, slip gradient, and the relative contribution of plastic strain to seismic moment. Peak slip velocity and rupture speed remain bounded. From fracture mechanics arguments, we derive a nonlinear relation between their limiting values, appropriate also for crack-like and supershear ruptures. At low background stress, plasticity turns self-similar pulses into steady state pulses, for which plastic strain contributes significantly to the seismic moment. We find that the closeness to failure of the background stress state is an adequate predictor of rupture speed for relatively slow events. Our proposed relations between state of stress and earthquake source properties in the presence of off-fault plasticity may contribute to the improved interpretation of earthquake observations and to pseudodynamic source modeling for ground motion prediction.