The relativistic Schrödinger theory (RST) for N-fermion systems is further elaborated with respect to three fundamental problems which must emerge in any relativistic theory of quantum matter: (i) emergence/suppression of exchange forces between identical/non-identical particles, (ii) self-interactions, (iii) non-relativistic approximation. These questions are studied in detail for two- and three-particle systems but the results do apply to a general N-particle system. As a concrete demonstration, the singlet and triplet configurations of the positronium groundstate are considered within the RST framework, including (...) a discussion of the corresponding hyperfine splitting. (shrink)