<p>The present investigation aims to uncover the analytical exact solutions within the context of Jacobi elliptic function periodic backgrounds of the complex mKdV equation with a time-varying coefficient, which may have potential physical applications in nonlinear optics and electrodynamics. By developing new eigenvalue solutions for the associated Lax pair and employing the well-established <i>N</i>-fold Darboux transformation, the breather and rogue wave solutions situated on the cn-Jacobi elliptic function background are analytically derived, whereas the breather and rational soliton solutions existing on the dn-Jacobi elliptic function background are rigorously given. By changing the time-varying coefficient, the dynamical behaviors of diverse localized wave structures are systematically analyzed and their spatial-temporal evolutions are graphically visualized. The spectral analysis of certain soliton and RS solutions are also discussed. These results provide valuable theoretical understanding of localized wave propagation in optics and electrodynamics, particularly in periodic media described by nonlinear equations with variable coefficients.</p>