In this paper we consider further applications of $(n,m)$-functions for the construction of 2-designs. For instance, we provide a new application of the extended Assmus-Mattson theorem, by showing that linear codes of APN functions with the classical Walsh spectrum support 2-designs. On the other hand, we use linear codes and combinatorial designs in order to study important properties of $(n,m)$-functions. In particular, we give a new design-theoretic characterization of $(n,m)$-plateaued and $(n,m)$-bent functions and provide a coding-theoretic as well as a design-theoretic interpretation of the extendability problem for $(n,m)$-bent functions.