We performed a radial velocity (RV) monitoring of the 40 Myr old star DS Tuc A with HARPS at the ESO-3.6m to determine the planetary mass of its 8.14-days planet, first revealed by TESS. We also observed two planetary transits with HARPS and ESPRESSO at ESO-VLT, to measure the Rossiter-McLaughlin (RM) effect and characterise the planetary atmosphere. We measured the high-energy emission of the host with XMM observations to investigate models for atmospheric evaporation. We employed Gaussian Processes (GP) regression to model the high level of the stellar activity, which is more than 40 times larger than the expected RV planetary signal. We extracted the transmission spectrum of DS Tuc A b from the ESPRESSO data and searched for atmospheric elements/molecules either by single-line retrieval and by performing cross-correlation with a set of theoretical templates. Through a set of simulations, we evaluated different scenarios for the atmospheric photo-evaporation of the planet induced by the strong XUV stellar irradiation. While the stellar activity prevented us from obtaining a clear detection of the planetary signal from the RVs, we set a robust mass upper limit of 14.4 M_e for DS Tuc A b. We also confirm that the planetary system is almost (but not perfectly) aligned. The strong level of stellar activity hampers the detection of any atmospheric compounds, in line with other studies presented in the literature. The expected evolution of DS Tuc A b from our grid of models indicates that the planetary radius after the photo-evaporation phase will fall within the Fulton gap. The comparison of the available parameters of known young transiting planets with the distribution of their mature counterpart confirms that the former are characterised by a low density, with DS Tuc A b being one of the less dense.
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