We theoretically investigate strong-coupling properties of an attractively interacting Fermi atomic gas, where the Cooper-pair formation occurs between atoms belonging to different dimensional bands. Including pairing fluctuations within the framework of the self-consistent $T$-matrix approximation (SCTMA), we examine how the BCS-type superfluid phase transition temperature $T_c$ varies as one moves from the 3D-3D to the 2D-3D system, in the wide parameter region with respect to the strength of the pairing interaction. In the 2D-3D limit, we find that, while the mean-field BCS theory predicts $T_c>0$ in the strong-coupling regime, $T_c$ is remarkably suppressed down to zero by pairing fluctuations that are strongly enhanced by the mixed-dimensionality of the system. As the origin of this, we clarify that the lower-dimensional (2D) component dominates the superfluid instability, so that the vanishing $T_c$ is the same phenomenon as that in the 2D-2D case. We also point out that this can already be seen in the mean-field level, when one examines the propagation of the Goldstone mode. On the other hand, we find that the pseudogap phenomenon, which is known as a precursor of Cooper-pair formation, exhibits a 3D character of the 2D-3D system. Thus, the effective dimensionality of the 2D-3D system depends on what we observe, reflecting the dimensional character of the elementary excitations that govern it.