We present a precise characterization of the TOI-561 planetary system obtained by combining previously published data with TESS and CHEOPS photometry, and a new set of 62 HARPS-N radial velocities (RVs). Our joint analysis confirms the presence of four transiting planets, namely TOI-561 b (P = 0.45 d, R = 1.42 R⊕, M = 2.0 M⊕), c (P = 10.78 d, R = 2.91 R⊕, M = 5.4 M⊕), d (P = 25.7 d, R = 2.82 R⊕, M = 13.2 M⊕), and e (P = 77 d, R = 2.55 R⊕, M = 12.6 R⊕). Moreover, we identify an additional, long-period signal (>450 d) in the RVs, which could be due to either an external planetary companion or to stellar magnetic activity. The precise masses and radii obtained for the four planets allowed us to conduct interior structure and atmospheric escape modelling. TOI-561 b is confirmed to be the lowest density (ρb = 3.8 ± 0.5 g cm-3) ultra-short period (USP) planet known to date, and the low metallicity of the host star makes it consistent with the general bulk density-stellar metallicity trend. According to our interior structure modelling, planet b has basically no gas envelope, and it could host a certain amount of water. In contrast, TOI-561 c, d, and e likely retained an H/He envelope, in addition to a possibly large water layer. The inferred planetary compositions suggest different atmospheric evolutionary paths, with planets b and c having experienced significant gas loss, and planets d and e showing an atmospheric content consistent with the original one. The uniqueness of the USP planet, the presence of the long-period planet TOI-561 e, and the complex architecture make this system an appealing target for follow-up studies.