Probing the physics of gamma-ray bursts through high-energy and multi-messenger observations

$\gamma$-ray bursts (GRBs) are among the most enigmatic and energetic phenomena of the Universe. After decades of theoretical studies and observational results, they still puzzle the astronomical community. Powered by a compact central engine, either a black hole or a neutron star, these objects are able to launch ultra-relativistic jets whose interaction with the surrounding medium leads to an electromagnetic emission visible from radio wavelengths up to very high-energy gamma rays. Besides being unique tools to probe relativistic astrophysics through multi-band observations, the detection of a $\gamma$-ray flash in coincidence with the binary neutron star merger GW170817 showed the groundbreaking potential of GRBs as multi-messenger sources, which are able to unveil properties of relativistic jets, nucleosynthesis of heavy elements, to evaluate the expansion rate of the Universe and set constraints on fundamental physics. However, despite the remarkable advances achieved in the last years, many open questions remain to be addressed, such as the origin of prompt emission, the composition and geometrical structure of relativistic jets, the nature of the emitting particles, the connection between the central engine and the afterglow physics, the acceleration mechanisms or the jet launching process. This thesis is devoted to understand the physics governing GRBs using $\gamma$/X-ray observations and to evaluate the perspectives opened by gravitational-wave observations. Exploiting the wealth of data provided in almost twenty years of activity of the Neil Gehrels \emph{Swift} Telescope, I investigated the spectral and temporal properties of GRB X-ray light curves. In particular, I analysed the transition phase between the prompt and the afterglow emission, which typically shows a steep flux decay phase, and I discovered a new relation between X-ray flux and spectral index. This relation challenges the common interpretation of the steep decay phase and requires the presence of specific radiative and cooling processes of the emitting particles. This study enabled a refined understanding of the prompt emission physics probing the jet nature, dynamics, structure and composition after the bulk of $\gamma$-ray radiation is released. In order to investigate the late time activity of the central engine, I also focused my analysis on the plateau phase of GRBs, whose origin is a longstanding open issue. After having defined a complete sample of GRBs with an X-ray plateau, a combined optical/X-ray time resolved spectral analysis of the plateau sample has been performed to probe the emission mechanism and discuss the main consequences for the currently available physical scenarios, which interpret the plateau as due to the presence of a central magnetar activity or the high latitude emission from a structured jet. The modelling developed to describe the GRB high-energy emission has then been used to evaluate the perspectives of multi-messenger astronomy with the next generation of gravitational-wave detectors, such as Einstein Telescope (ET) and $\gamma$/X-ray mission concepts, such as THESEUS. My thesis work provides a theoretical setup able to robustly simulate the statistical and observational properties of a population of short GRBs in the gravitational and electromagnetic domain. The versatility of our approach allowed us to apply this method for diverse purposes, such as evaluate the number of binary neutron star merger observed with an associated GRB for different ET configurations or the number of GRBs detectable by future high-energy mission concepts assuming different observational strategies. Synergies between the next GW detectors and high-energy satellites are a unique opportunity to detect, localise and characterise the electromagnetic counterpart of binary neutron star merger up to cosmological distances and shed light on GRB physics.