Using a population number synthesis code with detailed binary evolution, we calculate the distribution of the number of Type Ia supernovae as a function of time after starburst. This is done for both main progenitor scenarios (single degenerate and double degenerate), but also with various evolutionary assumptions (such as mass transfer efficiency, angular momentum loss, and common envelope description). The comparison of these theoretically predicted delay time distributions with observations in elliptical galaxies then allows us to constrain the evolutionary scenarios and parameters. From the morphological shape of the distributions, we conclude that all supernovae Ia cannot be produced through the single degenerate scenario alone, with the best match being obtained when both scenarios contribute. Within the double degenerate scenario, most systems go through a phase of quasi-conservative, stable Roche lobe overflow. We propose stellar rotation as a possible solution for the underestimation of the absolute numbers of events predicted by many theoretical population synthesis studies. A brief comparison with these other studies is made, showing good correspondence under the nontrivial condition of equivalent assumptions. We also investigate the influence of different supernova Ia progenitors and evolutionary parameters on the theoretical distribution of the iron abundance of G-type dwarfs in the Galactic disk. These stars are good indicators of the entire chemical history of the Galaxy, and their predicted metallicity distribution can also be compared to the observed ones. This again limits the number of acceptable combinations of assumptions. Supporting previous results, the best correspondence is found in the case where both the single and double degenerate scenario contribute.