In this paper, we investigate the relationship between free hypermodules and normal projective hypermodules. In particular, we prove that every cyclic free $R$-hypermodule is normal projective and establish a criterion characterizing the normal projectivity of subhypermodules of the free hypermodule $H(R)$, where $R$ is a hyperring. In addition, by means of quotient hypermodule constructions, we provide an example of a free hypermodule arising from a free module. Furthermore, we study the normal injectivity of simple hypermodules and obtain a fundamental characterization of hyperrings whose simple hypermodules are normal injective.