Despite being the most abundant classes of known exoplanets, super-Earths and mini-Neptunes are absent in our Solar System, and the exact reason for this is still unknown. Some models suggest that cold-Jupiters play a major role in the formation of inner low-mass planets. This could explain the current architecture of our Solar System, and the reason for the absence of any super-Earth in close orbit around the Sun. In this framework, we present the combination of Radial Velocity (RV) and astrometry as a comprehensive approach to the study of hierarchical systems, hosting both long- and short-period planetary companions. Taking advantage of high-resolution echelle spectroscopy from HARPS-N, and high-precision Gaia astrometry, we study planetary systems across different spectral types, to disclose the dependence of the formation of multi-planet systems on the mass of the host star. We present the high-cadence and high-precision RV monitoring of stellar hosts to long-period giants with well-measured orbits, in search for short-period low-mass planets, over different spectral types from M to G. We then show how the combination of RV measurements and high-precision Gaia astrometry can greatly expand our knowledge of the long-period planets in these systems: combining RVs with proper-motion anomalies from Gaia and Hipparcos, we are able to measure the inclination and real masses of several long-period companions in our sample. Moreover, with a set of simulations of Gaia DR4 astrometric time series, we quantify the great boost in precision that these data will bring to the study of long-period exoplanetary companions. Finally, from an homogeneous Bayesian analysis of the observed sample, we highlight a very different behavior between G- and M-dwarfs: in the G-sample, the frequency of mini-Neptunes appears to be lower in the presence of cold Jupiters, while the opposite is observed for M dwarfs. These results provide an important advancement in discriminating between the different formation models of sub-Neptunes in the presence of outer giant companions, and produce the first direct observation of a different influence of cold Jupiters in the formation of planetary systems around stars of different masses.