Context. Giant planets in short-period orbits around bright stars represent optimal candidates for atmospheric and dynamical studies of exoplanetary systems. Aims. We aim to analyse four transits of WASP-33b observed with the optical high-resolution HARPS-N spectrograph to confirm its nodal precession, study its atmosphere, and investigate the presence of star-planet interactions. Methods. We extracted the mean line profiles of the spectra using the least-squares deconvolution method, and we analysed the Doppler shadow and the radial velocities. We also derived the transmission spectrum of the planet, correcting it for the stellar contamination due to rotation, centre-to-limb variations, and pulsations. Results. We confirm the previously discovered nodal precession of WASP-33b, almost doubling the time coverage of the inclination and projected spin-orbit angle variation. We find that the projected obliquity reached a minimum in 2011, and we used this constraint to derive the geometry of the system, and in particular its obliquity at that epoch (ϵ = 113.99° ± 0.22°) and the inclination of the stellar spin axis (is = 90.11° ± 0.12°). We also derived the gravitational quadrupole moment of the star J2 = (6.73 ± 0.22) × 10−5, which we find to be in close agreement with the theoretically predicted value. Small systematics errors are computed by shifting the date of the minimum projected obliquity. We present detections of Hα and Hβ absorption in the atmosphere of the planet, with a contrast almost twice as small as that previously detected in the literature. We also find evidence for the presence of a pre-transit signal, which repeats in all four analysed transits and should thus be related to the planet. The most likely explanation lies in a possible excitation of a stellar pulsation mode by the presence of the planetary companion. Conclusions. A future common analysis of all available datasets in the literature will help shed light on the possibility that the observed Balmer lines’ transit depth variations are related to stellar activity and pulsation, and to set constraints on the planetary temperature–pressure structure and thus on the energetics possibly driving atmospheric escape. A complete orbital phase coverage of WASP-33b with high-resolution spectroscopic (and spectro-polarimetric) observations could help us to understand the nature of the pre-transit signal.