Coherent interactions between free electrons and optical fields offer a promising route to quantum control, nonclassical state generation, and programmable quantum dynamics. At kiloelectron-volt electron energies, the emission or absorption of optical photons produces appreciable quantum recoil, resolving the electron into a discrete and controllable energy ladder. Starting from relativistic quantum electrodynamics, we derive an exact recoil-resolved interaction Hamiltonian in a traveling-wave picture. The resulting recoil ladder forms a high-dimensional qudit with programmable couplings and sufficient controllability for universal quantum computation. We demonstrate its use for analogue quantum simulation, including one-dimensional curved-spacetime models and synthetic black-hole horizons, as well as for quantum information processing, where multiple logical qubits and high-fidelity gates can be realized within a single electron. In parallel, the same recoil-enabled dynamics allow the controlled creation of complex hybrid electron–photon states, in which engineered ladder transitions imprint nonclassical correlations and structure onto the emitted light. Together, these results establish recoiled free electrons as a versatile platform bridging quantum optics, quantum control, Hamiltonian engineering, quantum simulation, and quantum information processing.