<p>Water waves play a crucial role in elucidating fluid dynamics and plasma physics, as they significantly influence phenomena such as wave propagation, energy transfer, and stability across diverse environments. In this paper, we aim to construct the bilinear auto-Bäcklund transformation and novel function superposition solutions for an extended (3+1)-dimensional generalized shallow water wave equation by employing the Hirota bilinear method and symbolic computation. We meticulously illustrate and analyze the dynamic behavior of interactions between lump waves and two kink waves under various periodic wave backgrounds. This includes the collision of the lump wave with the two kink waves, the splitting of the kink waves, and the emergence and degeneration of the lump wave. By scrutinizing the interactions of breather waves, we observe the fusion and separation phenomena involving bell-shaped waves and breather waves. Additionally, by examining the interactions of rogue waves, we analyze their fission and fusion processes. The interplay among these waves can significantly enhance our understanding of the characteristics of solutions involving function superposition, potentially shedding light on certain physical phenomena within the realm of nonlinear science.</p>