Compact objects, such as Neutron stars (NSs) and black holes (BHs), are the remnants of massive stars and play a central role in some of the most energetic phenomena in the Universe. Through accretion and merger processes, they power X-ray binaries and gravitational-wave (GW) sources, respectively. These systems are key to probing the physics of extreme gravity, dense matter, and relativistic astrophysics. The ever-growing samples from GW observations provide an unprecedented opportunity to advance our understanding of stellar and binary evolution through population studies in the era of multi-messenger astronomy.

This thesis presents binary population synthesis (BPS) studies of compact-object binaries containing BHs and NSs, focusing on wind-fed BH high-mass X-ray binaries and merging NS--BH (NSBH) systems. Through these studies, we investigate several open questions, including the origin of BH spins, distinguishable channel-wise characteristics of merging NSBHs, and the existence of lower mass-gap BHs.

These populations are investigated using the next-generation BPS code POSYDON, which combines detailed binary evolution models, computed using the stellar evolution code MESA, with a self-consistent population framework. It includes full stellar structure, rotation, and realistic modeling of mass transfer, enabling more physically accurate simulations of binary interactions and compact object formation.

The results of this thesis highlight that accurately modeling key physical processes, using physically motivated approaches is essential for interpreting current and future multi-messenger observations, and for deepening our understanding of stellar and binary evolution. As gravitational-wave and electromagnetic observations continue to grow, their integration with advanced BPS studies will be crucial for probing the underlying physics of compact object binaries. This thesis establishes a foundation for more reliable and unified population modeling in the era of multi-messenger astronomy.