We use stripped-down versions of three semi-analytic galaxy formation models to study the influence of different assumptions about gas cooling and galaxy mergers. By running the three models on identical sets of merger trees extracted from high-resolution cosmological N-body simulations, we are able to perform both statistical analyses and halo-by-halo comparisons. This paper demonstrates that there is a good statistical agreement between the three models used here, when operating on the same merger trees, reflecting a general agreement in the underlying framework for semi-analytic models. We also show, however, that various assumptions that are commonly adopted to treat gas cooling and galaxy mergers can lead to significantly different results, at least in some regimes. In particular, we find that the different models adopted for gas cooling lead to similar results for mass scales comparable to that of our own Galaxy. Significant differences, however, arise at larger mass scales. These are largely (but not entirely) due to different treatments of the ‘rapid cooling’ regime, and different assumptions about the hot gas distribution. At this mass regime, the predicted cooling rates can differ up to about one order of magnitude, with important implications on the relative weight that these models give to feedback from active galactic nuclei in order to counteract excessive gas condensation in relatively massive haloes at low redshift. Different assumptions in the modelling of galaxy mergers can also result in significant differences in the timings of mergers, with important consequences for the formation and evolution of massive galaxies.