The proximity of a transition metal dichalcogenide (TMD) to graphene imprints a rich spin texture in graphene and complements its high quality charge/spin transport by inducing spin-orbit coupling (SOC). Rashba and valley-Zeeman SOCs are the origin of charge-to-spin conversion mechanisms such as Rashba-Edelstein effect (REE) and spin Hall effect (SHE). In this work, we experimentally demonstrate for the first time charge-to-spin conversion due to the REE in a monolayer WS2-graphene van der Waals heterostructure. We measure the current-induced spin polarization up to room temperature and control it by a gate electric field. Our observation of the REE and inverse of the effect (IREE) is accompanied by the SHE which we discriminate by symmetry-resolved spin precession under oblique magnetic fields. These measurements also allow for quantification of the efficiencies of charge-to-spin conversion by each of the two effects. These findings are a clear indication of induced Rashba and valley-Zeeman SOC in graphene that lead to generation of spin accumulation and spin current without using ferromagnetic electrodes. These realizations have considerable significance for spintronic applications, providing accessible routes towards all-electrical spin generation and manipulation in two-dimensional materials.