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 res...