We present p-gadget3-k, an updated version of gadget-3, that incorporates the chemistry package krome. p-gadget3-k follows the hydrodynamical and chemical evolution of cosmic structures, incorporating the chemistry and cooling of H-2 and metal cooling in non-equilibrium. We performed different runs of the same ICs to assess the impact of various physical parameters and prescriptions, namely gas metallicity, molecular hydrogen formation on dust, star formation recipes including or not H-2 dependence, and the effects of numerical resolution. We find that the characteristics of the simulated systems, both globally and at kpc-scales, are in good agreement with several observable properties of molecular gas in star-forming galaxies. The surface density profiles of star formation rate (SFR) and H-2 are found to vary with the clumping factor and resolution. In agreement with previous results, the chemical enrichment of the gas component is found to be a key ingredient to model the formation and distribution of H-2 as a function of gas density and temperature. A star formation algorithm that takes into account the H-2 fraction together with a treatment for the local stellar radiation field improves the agreement with observed H-2 abundances over a wide range of gas densities and with the molecular Kennicutt-Schmidt law, implying a more realistic modelling of the star formation process.