The spin-orbit and coherent couplings along with rotation can combine together to result in scalar potentials which can modify the trapping potential to yield a variety of effective potentials experienced by a particle. The bosons in these rotating SO- and coherently-coupled Bose-Einstein condensates (BECs) can thus be subjected to rotating effective potentials equivalent to a toroidal, a symmetric double-well, an asymmetric double-well potentials, etc. With the increase in rotation frequency, the magnitude of spin-expectation per particle for even an antiferromagnetic SO-coupled BEC tends to unity, which is consistent with our analysis of the eigenfunctions and eigenenergies of the non-interacting Hamiltonian. The breakdown in the degeneracy of the minimum energy eigenfunctions of the non-interacting Hamiltonian results in the similar behavior of ferromagnetic and antiferromagnetic SO-coupled BECs. This is illustrated by the similar static vortex-lattice patterns, associated spin-textures, and mass-currents for the two systems at fast rotations.