Context. Mass transfer stability is a key issue in studies of binary evolution. Critical mass ratios for dynamically stable mass transfer have been analyzed on the basis of an adiabatic mass loss model, finding that the donor stars on the giant branches tend to be more stable than that based on the composite polytropic stellar model. Double white dwarfs (DWDs) are of great importance in many fields and their properties would be significantly affected under the new mass transfer stability criterion. Aims. We seek to investigate the influence of mass transfer stability on the formation and properties of DWD populations and discuss the implications in supernova Type Ia (SN Ia) and gravitational wave (GW) sources. Methods. We performed a series of binary population synthesis, adopting the critical mass ratios from the adiabatic mass loss model (i.e., Ge's model) and that of the composite polytropic model, respectively. In each simulation, 5×106 binaries were included and evolved from zero-age main sequence to the end of their evolution and the DWDs were gradually obtained. Results. For Ge's model, most of the DWDs are produced from the stable non-conservative Roche lobe (RL) overflow, along with a common-envelope (CE) ejection channel (RL+CE channel), regardless of the CE ejection efficiency, αCE. Conversely, the results of the polytropic model strongly depend on the adopted value of αCE. We find DWDs produced from the RL+CE channel have comparable WD masses and the mass ratio distribution peaks at around 1. Based on the magnitude-limited sample of DWDs, the space densities for the detectable DWDs and those with extremely low-mass WD (ELM WD) companions in Ge's model is: 1347 kpc-3 and 473 kpc-3, respectively, which is close to what has been shown in observations. On the other hand, the polytropic model overpredicts space density of DWDs by a factor of about 2-3. We also find that the results of DWD merger rate distribution per Galaxy in Ge's model reproduce the observations better than that of the polytropic model, and the merger rate of DWDs with ELM WD companions in the Galaxy is about 1.8×10-3 yr-1 in Ge's model. This result is comparable to the observation estimation of 2×10-3 yr-1. The findings from Ge's model predict a Galactic SN Ia rate of ~6×10-3 yr-1 from DWDs, supporting observations of (5.4 ± 1.2) × 10-3 yr-1. For the fiducial model of αCE αCE =1, the number of detectable GW sources in the polytropic model is larger than that in Ge's model by about 35%. Conclusions. We confirm that mass transfer stability plays an important role in the formation and properties of DWD populations as well as in the progenitors of SNe Ia and detectable GW sources. The results of Ge's model support the observational DWD merger rate distribution per Galaxy and the space density of DWDs in the Galaxy. © The Authors 2023.