Super-Earths and sub-Neptunes are the most common types of planets outside the Solar System and likely represent the link between terrestrial planets and gas giants. Characterizing their physical and orbital properties and studying their multiplicity are key steps in testing and understanding their formation, migration, and evolution. We examined the star BD+00 444 (GJ 105.5, TOI-2443; V=9.5mag; d=23.9pc) in depth, with the aim of characterizing and confirming the planetary nature of its small companion, the planet candidate TOI-2443.01, which was discovered by the TESS space telescope and subsequently validated by a follow-up statistical study. We monitored BD+00 444 with the HARPS-N spectrograph for 1.5 years to search for planet-induced radial-velocity (RV) variations, and then analyzed the RV measurements jointly with TESS and ground-based photometry. We determined that the host is a quiet K5 V star with a radius of R*=0.631_-0.014_^+0.013^R_{sun}_ and a mass of M*=0.642_-0.025_^+0.026^M_{sun}_. We revealed that the sub-Neptune BD+00 444 b has a radius of Rb=2.36+/-0.05R_{Earth}_, a mass of Mb=4.8+/-1.1M_{Earth}_, and consequently a rather low-density value of {rho}b=2.00_-0.45_^+0.49^g/cm3, which makes it compatible with both an Earth-like rocky interior with a thin H-He atmosphere and a half-rocky, half-water composition with a small amount of H-He. With an orbital period of about 15.67 days and an equilibrium temperature of about 519K, BD+00 444 b has an estimated transmission spectroscopy metric (TSM) of 159_-31_^+46^, which makes it ideal for atmospheric follow-up with the James Webb Space Telescope. Notably, it is the second most eccentric inner transiting planet among those with well-determined eccentricities, with e=0.302_-0.035_^+0.051^, and a mass of below 20M_{Earth}_. We estimated that tidal forces from the host star affect both the rotation and eccentricity of planet b, and strong tidal dissipation may signal intense volcanic activity. Furthermore, our analysis suggests the presence of a sub-Neptune-mass planet candidate, BD+00 444 c, which would have an orbital period of Pc=96.6+/-1.4 days and a minimum mass of Mcsini=9.3_-2.0_^+1.8^M_{Earth}_. With an equilibrium temperature of about 283K, BD+00 444 c is inside the habitable zone; however, confirmation of this candidate would require further observations and stronger statistical evidence. We explored the formation and migration of both planets by means of population synthesis models, which reveal that both planets started their formation beyond the water snowline during the earliest phases of the life of their protoplanetary disk.