A novel coexisting passive reconfigurable intelligent surface (RIS) and active decode-and-forward (DF) relay assisted non-orthogonal multiple access (NOMA) transmission framework is proposed. In particular, two communication protocols are conceived, namely Hybrid NOMA (H-NOMA) and Full NOMA (F-NOMA). Based on the proposed two protocols, both the sum rate maximization and max-min rate fairness problems are formulated for jointly optimizing the power allocation at the access point and relay as well as the passive beamforming design at the RIS. To tackle the non-convex problems, an alternating optimization (AO) based algorithm is first developed, where the transmit power and the RIS phase-shift are alternatingly optimized by leveraging the two-dimensional search and rank-relaxed difference-of-convex (DC) programming, respectively. Then, a two-layer penalty based joint optimization (JO) algorithm is developed to jointly optimize the resource allocation coefficients within each iteration. Finally, numerical results demonstrate that: i) the proposed coexisting RIS and relay assisted transmission framework is capable of achieving a significant user performance improvement than conventional schemes without RIS or relay; ii) compared with the AO algorithm, the JO algorithm requires less execution time at the cost of a slight performance loss; and iii) the H-NOMA and F-NOMA protocols are generally preferable for ensuring user rate fairness and enhancing user sum rate, respectively.