Heterostructures containing strongly correlated electron systems provide a platform to clarify interplay of electron correlation and Rashba spin-orbit coupling in unconventional superconductors. Motivated by recent fabrication of artificially engineered heavy fermion superlattices and high-temperature cuprate superconductors, we conduct a thorough study on superconductivity in the Rashba-Hubbard model. In contrast to previous weak coupling approaches, we employ fluctuation-exchange approximation to describe quantum critical magnetic fluctuations and resulting superconductivity. As a result, robust Fermi surfaces against magnetic fluctuations, incommensurate spin fluctuations, and a strongly parity-mixed superconducting phase are demonstrated in a wide range of electron filling from type-II van Hove singularity to half filling. We also clarify impacts of type-II van Hove singularity on magnetic fluctuations and superconductivity. Whereas the d(x2-y2) -wave pairing is always dominant, subdominant spin-triplet pairing with either p-wave or f-wave symmetry shows a comparable magnitude, especially near the type-II van Hove singularity. Our results resolve unsettled issues on strongly correlated Rashba systems and uncover candidate systems of nonreciprocal transport and topological superconductivity.