<p>In this paper, we carry out a systematic study of the coupled cubic-quintic nonlinear Schrödinger equations, which describe the effects of quintic nonlinearity on the ultrashort optical pulse propagation in non-Kerr media. By applying the Darboux transformation, we attain the interactions of different types of breathers and rogue waves on the periodic background, some of which are novel. It is worth noting that changes in the values of parameters affect the interaction properties, structures and energy conversion of these solutions. In particular, we categorize these solutions and describe the precise conditions under which they arise, making a unique contribution to the study of interactions between novel localized waves on the periodic background. We hope that the work we have done will contribute to making it easier to generate other novel localized waves on various backgrounds for other integrable systems.</p>