We consider the ground-state properties of Rashba spin-orbit-coupled pseudo-spin-1/2 Bose-Einstein condensates (BECs) in a rotating two-dimensional (2D) toroidal trap. In the absence of spin-orbit coupling (SOC), the increasing rotation frequency enhances the creation of giant vortices for the initially miscible BECs, while it can lead to the formation of semiring density patterns with irregular hidden vortex structures for the initially immiscible BECs. Without rotation, strong 2D isotropic SOC yields a heliciform-stripe phase for the initially immiscible BECs. Combined effects of rotation, SOC, and interatomic interactions on the vortex structures and typical spin textures of the ground state of the system are discussed systematically. In particular, for fixed rotation frequency above the critical value, the increasing isotropic SOC favors a visible vortex ring in each component which is accompanied by a hidden giant vortex plus a (several) hidden vortex ring(s) in the central region. In the case of 1D anisotropic SOC, large SOC strength results in the generation of hidden linear vortex string and the transition from initial phase separation (phase mixing) to phase mixing (phase separation). Furthermore, the peculiar spin textures including skyrmion lattice, skyrmion pair and skyrmion string are revealed in this system.