The intermetallic compound δ-Ni₅Ga₃ has emerged as a promising catalyst for CO₂ hydrogenation to methanol, offering high selectivity at low-pressure operation, and enhanced stability compared to conventional Cu/ZnO catalysts. However, the fundamental understanding of its active sites, reaction mechanisms, and deactivation pathways remains incomplete, hindering its further development. In this study, we utilize well-defined δ-Ni₅Ga₃ thin film model catalysts synthesized via magnetron sputtering to investigate these aspects under realistic reaction conditions. We investigate the evolution of the catalyst with temperature employing in situ ambient pressure X-ray photoelectron spectroscopy (AP-XPS) at 300 mbar, microreactor activity measurements, temperature-programmed desorption (TPD), and density functional theory (DFT) calculations. Our experiments show the active catalyst as mostly metallic with only small amounts on oxidized gallium, which gradually reduces and gives way to an increased nickel-concentration at the surface at higher temperatures, accompanied by carbide-growth. We further observe the temperature-evolution of key intermediates, such as carboxyl, formate, and methoxy species. Based on these observations, we discuss distinct pathways for methanol synthesis and CO₂ methanation, with methoxy formation correlating directly with methanol activity, as well as the deactivation mechanism.