Context. The analysis of galaxy properties, such as stellar masses, colors, sizes and morphologies, and the relations among them and the environment, in which the galaxies reside, can be used to investigate the physical processes driving galaxy evolution. Aims. We conduct a thorough study of the cluster A209 with a new large spectro-photometric dataset to investigate possible environmental effects on galaxy properties that can provide information on galaxy evolution in cluster hostile environments. Methods. We use the dataset obtained as part of the CLASH-VLT spectroscopic survey, supplemented with Subaru/SuprimeCam high-quality imaging in BVRIz-bands, which yields 1916 cluster members (50% of them spectroscopically confirmed) down to a stellar mass M∗ = 108.6 M⊙. We determine the stellar mass function of these galaxies in different regions of the cluster, by separating the sample into star-forming and passive cluster members. We then determine the intra-cluster light and its properties. We also derive the orbits of low- (M∗ ≤ 1010.0 M⊙) and high-mass (M∗ > 1010.0 M⊙) passive galaxies and study the effect of the environment on the mass-size relation of early-type galaxies, selected according to their Sérsic index; the effects are studied separately for the galaxies in each mass range. Finally, we compare the cluster stellar mass density profile with the number density and total-mass density profiles. Results. The stellar mass function of the star-forming cluster galaxies does not depend on the environment. The slope found for passive galaxies becomes flatter in the densest cluster region, which implies that the low-mass component starts to dominate when moving away from the cluster center. The color distribution of the intra-cluster light is consistent with the color of passive cluster members. The analysis of the dynamical orbits of passive galaxies shows that low-mass galaxies have tangential orbits, avoiding small pericenters around the BCG. The mass-size relation of low-mass passive early-type galaxies is flatter than that of high-mass galaxies, and its slope is consistent with the slope of the relation of field star-forming galaxies. Low-mass galaxies are also more compact within the scale radius of 0.65 Mpc. The ratio between the stellar and number density profiles shows a mass segregation effect in the cluster center. The comparative analysis of the stellar and total density profiles indicates that this effect is due to dynamical friction. Conclusions. Our results are consistent with a scenario in which the "environmental quenching" of low-mass galaxies is due to mechanisms such as harassment out to r200, starvation, and ram-pressure stripping at smaller radii. This scenario is supported by the analysis of the mass function, of the dynamical orbits and of the mass-size relation of passive early-type galaxies in different cluster regions. Moreover, our analyses support the idea that the intra-cluster light is formed through the tidal disruption of subgiant (M∗ ∼ 109.5-10.0 M⊙) galaxies. In fact, our results suggest that low-mass galaxies are destroyed by tidal interactions, and that those that avoid small pericenters around the BCG are influenced by tidal interactions that reduce their sizes. We suggest dynamical friction as the process responsible for the observed mass segregation.