For linear systems with time-varying input delay and additive disturbances we show that the basic predictor feedback control law is inverse optimal, with respect to a meaningful differential game problem, and establish its robustness to constant multiplicative perturbations appearing at the system input. Both of these properties of the basic predictor feedback controller have not been established so far, even for the constant-delay case. We then show that the basic predictor feedback controller, when applied through a low-pass filter, is again inverse optimal and study its input-to-state stability as well as its robustness, to the low-pass filter time constant properties. All of the stability and inverse optimality proofs are based on the infinite-dimensional backstepping transformation, which allows us to construct appropriate Lyapunov functionals. A numerical example is also provided.