We study two-dimensional characteristic free-boundary problems in ideal compressible magnetohydrodynamics. For current-vortex sheets, surface-wave effects yield derivative loss and only weak (neutral) stability; under a sufficient stability condition on the background state we obtain anisotropic weighted Sobolev energy estimates and prove local-in-time existence and nonlinear stability via a Nash-Moser scheme, confirming stabilization by strong magnetic fields against Kelvin-Helmholtz instability. For the plasma-vacuum interface, coupling hyperbolic MHD with elliptic pre-Maxwell dynamics, we establish local existence and uniqueness provided at least one magnetic field is nonzero along the initial interface.