Low-Earth-orbit (LEO) satellite communication systems face challenges due to high satellite mobility, which hinders the reliable acquisition of instantaneous channel state information at the transmitter (CSIT) and subsequently degrades multi-user transmission performance. This paper investigates a downlink multi-user multi-antenna system, and tackles the above challenges by introducing orthogonal time frequency space (OTFS) modulation and rate-splitting multiple access (RSMA) transmission. Specifically, OTFS enables stable characterization of time-varying channels by representing them in the delay-Doppler domain. However, realistic propagation introduces various inter-symbol and inter-user interference due to non-orthogonal yet practical rectangular pulse shaping, fractional delays, Doppler shifts, and imperfect (statistical) CSIT. In this context, RSMA offers promising robustness for interference mitigation and CSIT imperfections, and hence is integrated with OTFS to provide a comprehensive solution. A compact cross-domain input-output relationship for RSMA-OTFS is established, and an ergodic sum-rate maximization problem is formulated and solved using a weighted minimum mean-square-error based alternating optimization algorithm that does not depend on channel sparsity. Simulation results reveal that the considered practical propagation effects significantly degrade performance if unaddressed. Furthermore, the RSMA-OTFS scheme demonstrates improved ergodic sum-rate and robustness against CSIT uncertainty across various user deployments and CSIT qualities.