Toeplitz and Hankel operators are among the most fundamental classes of operators on Hilbert spaces, especially on Hardy and Bergman spaces. In recent years, these operators have been investigated in the setting of Fréchet spaces, including spaces of holomorphic functions on the unit disk, the entire complex plane, and power series spaces. In this talk, we investigate Toeplitz and Hankel operators acting on Köthe spaces. We focus on their operator-theoretic properties, including continuity and compactness, as well as their behavior under iteration. In particular, we study conditions related to topologizability and power boundedness. Building on these, we provide conditions for the Cesàro boundedness and mean ergodicity of the operators.