In this paper, we first determine the weight distributions of some subfield codes ${\mathcal{C}}_f^{(p)}$ and punctured codes ${\mathcal{C}}_f^{\ast (p)}$ constructed from special functions, both of which are few-weight codes. Furthermore, we derive the parameters of their dual codes. Notably, some of these codes and their duals turn out to be optimal or almost distance-optimal. As an application, two classes of 2-designs are constructed from some codes.