  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.
