The interstellar medium (ISM) is typically a hostile environment: cold, dilute and irradiated. Nevertheless, it appears very fertile for molecules. The localized heating resulting from turbulence dissipation is a possible channel to produce and excite molecules. However, large-scale simulations cannot resolve the dissipative scales of the ISM. Here, we present two-dimensional small-scale simulations of decaying hydrodynamic turbulence using the chemses code, with fully resolved viscous dissipation, time-dependent heating, cooling, chemistry and excitation of a few rotational levels of H2. We show that molecules are produced and excited in the wake of strong dissipation ridges. We carefully identify shocks and we assess their statistics and contribution to the molecular yields and excitation. We find that the formation of molecules is strongly linked to increased density as a result of shock compression and to the opening of endothermic chemical routes because of higher temperatures. We identify a new channel for molecule production via H2 excitation, illustrated by CH+ yields in our simulations. Despite low temperatures and the absence of magnetic fields (favouring CH+ production through ion-neutral velocity drifts), the excitation of the first few rotational levels of H2 shrinks the energy gap to form CH+. The present study demonstrates how dissipative chemistry can be modelled by statistical collections of one-dimensional steady-state shocks. Thus, the excitation of higher J levels of H2 is likely to be a direct signature of turbulence dissipation, and an indirect probe for molecule formation. We hope these results will help to bring new tools and ideas for the interpretation of current observations of H2 rotational lines carried out using the Stratospheric Observatory for Infrared Astronomy (SOFIA), and pave the way for a better understanding of the high-resolution mapping of H2 emission by future instruments, such as theJames Webb Space Telescope and the Space Infrared Telescope for Cosmology and Astrophysics.