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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) and the Tully-Fisher relation (TFR). We perform a morpho-kinematic modelling of 593 [Oii] emitters in various environments in the COSMOS area from the MUSE-gAlaxy Groups In Cosmos (MAGIC) 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]λλ3727, 3729 doublet to extract the ionised gas kinematic maps from the MUSE cubes, and we model them for a sample of 146 [Oii] emitters, with bulge and disk components constrained from morphology and a dark matter halo. We find an offset of 0.03 dex on the size-mass relation zero point between the field and the large structure subsamples, with a richness threshold of N=10 to separate between small and large structures, and of 0.06 dex with N=20. Similarly, we find a 0.1 dex difference on the MS with N=10 and 0.15 dex 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 0.04 dex. We discard the effect of quenching for the largest structures that 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, for roughly 0.7-1.5 Gyr. This result holds when including the gas mass, but vanishes once we include the asymmetric drift correction.