We use optical tweezers microrheology and fluorescence microscopy to apply nonlinear microscale strains to entangled and cross-linked actin networks, and measure the resulting stress and actin filament deformations. We couple nonlinear stress response and relaxation to the velocities and displacements of individual fluorescent-labeled actin segments, at varying times throughout the strain and varying distances from the strain path, to determine the underlying molecular dynamics that give rise to the debated nonlinear response and stress propagation of cross-linked and entangled actin networks at the microscale. We show that initial stress stiffening arises from acceleration of strained filaments due to molecular extension along the strain, while softening and yielding is coupled to filament deceleration, halting, and recoil. We also demonstrate a surprising nonmonotonic dependence of filament deformation on cross-linker concentration. Namely, networks with no cross-links or substantial cross-links both exhibit fast initial filament velocities and reduced molecular recoil while intermediate cross-linker concentrations display reduced velocities and increased recoil. We show that these collective results are due to a balance of network elasticity and force-induced cross-linker unbinding and rebinding. We further show that cross-links dominate entanglement dynamics when the length between cross-linkers becomes smaller than the length between entanglements. In accord with recent simulations, we demonstrate that post-strain stress can be long-lived in cross-linked networks by distributing stress to a small fraction of highly strained connected filaments that span the network and sustain the load, thereby allowing the rest of the network to recoil and relax.