Stars play a central role in modern astrophysics. To understand how stars form is therefore fundamental for the field. Stars form in gas clouds in the interstellar medium. This medium is magnetized and turbulent; star formation is therefore a complex, highly non-linear and multiscale problem. In this context, star formation processes are still not well-known, and particularly protostellar disk and multiple stars -- which means stars that are gravitationaly bound -- formation. Numerical simulations are crucial to have a better insight of these processes. This work is divided in two main parts, dedicated to the study of the first phases of star formation. The first part presents the numerical simulations I performed during my thesis to investigate protostellar disks and multiple star system formation. When the magnetic intensity is strong enough, it could prevent both protostellar disks formation and fragmentation through an efficient angular momentum transport process. I first develop an analytical and numerical study to show the key role played by the geometry of the collapse on the angular momentum transport processes. When the rotation axis of the prestellar core and the magnetic field are misaligned, magnetic braking is less efficient and massive disks formation is restored. The influence of turbulence on magnetic diffusion, disk formation, fragmentation and outflows -- one of the most important tracers of star formation -- is then discussed. Turbulence is responsible for an efficient magnetic diffusion in the central regions of the collapsing core as well as for a misalignment between the rotation axis and the magnetic field. Magnetic braking is thus efficiently reduced, and both massive disks formation and fragmentation may happen. The second part is dedicated to synthetic observations performed from our numerical simulations. Three different types of synthetic observations were performed: column-density maps, spectral energy distributions and visibilities. Following a classic analysis performed in observational studies, these observations will be compared to analytical models to try to deduce disk properties.
