The evolution of galaxies is influenced by many physical processes, which may vary depending on their environment. We combine Hubble Space Telescope (HST) and Multi-Unit Spectroscopic Explorer (MUSE) data of galaxies at 0.25<~z<~1.5 to probe the impact of environment on the size-mass relation, the main sequence (MS) relation, and the Tully-Fisher relation (TFR). We perform a morpho-kinematics modelling of 593 [OII] emitters in various environments in the COSMOS area from the MUSE-gAlaxy Groups In Cosmos survey. The HST F814W images are modelled with a bulge-disk decomposition to estimate their bulge-disk ratio, effective radius, and disk inclination. We use the [OII]{lambda}{lambda}3727,3729 doublet to extract the galaxies' ionised gas kinematics maps from the MUSE cubes, and we model those maps for a sample of 146 [OII] emitters, including bulge and disk components constrained from morphology and a dark matter halo. We find an offset of 0.03dex (1{sigma} significant) on the size-mass relation zero point between the field and the large structure sub-samples, with a richness threshold of N=10 to separate between small and large structures, and of 0.06dex (2{sigma}) with N=20. Similarly, we find a 0.1dex (2{sigma}) difference on the MS relation with N=10 and 0.15dex (3{sigma}) with N=20. These results suggest that galaxies in massive structures are smaller by 14% and have star formation rates reduced by a factor of 1.3-1.5 with respect to field galaxies at z=~0.7. Finally, we do not find any impact of the environment on the TFR, except when using N=20 with an offset of 0.04dex (1{sigma}). We discard the effect of quenching for the largest structures, which would lead to an offset in the opposite direction. We find that, at z=~0.7, if quenching impacts the mass budget of galaxies in structures, these galaxies would have been affected quite recently and for roughly 0.7-1.5Gyr. This result holds when including the gas mass but vanishes once we include the asymmetric drift correction.