<p>This study explores the stability and energy behavior of dust acoustic waves (DAWs) in a magnetized dusty plasma featuring electrons with a hybrid Cairns-Tsallis distribution, nonthermal ions, and negatively charged dust grains of variable sizes following a power-law distribution. The primary goal is to understand how different plasma settings affect the properties of these waves, including their amplitude, width, energy, and stability. By deriving the Zakharov-Kuznetsov (ZK) equation through the reductive perturbation method and conducting a stability analysis using the small-k expansion perturbation technique, we examine how parameters such as the nonextensive parameter, the nonthermal parameter, the dust radius ratio, and the power-law index affect the stability and energy of DAWs. Comparisons between systems with mono-sized and multi-sized dust grains show that multi-sized grains lead to higher phase velocities, larger amplitudes, and broader widths of solitary waves, with higher energy than mono-sized grains. The instability growth rate exhibits distinct behavior for mono-sized and multi-sized grains, particularly under varying magnetic fields and propagation angles. The results reveal that the wave energy and stability growth rates are significantly impacted by the plasma’s composition and parameter variations. Notably, certain parameters tend to increase wave energy and instability growth, while others suppress them. These findings highlight the crucial role of dust size distribution and electron distribution characteristics in shaping the nonlinear dynamics of DAWs, offering insights into dusty plasma environments in space, astrophysical and laboratory plasmas like those in cometary environments, Saturn’s F and G rings, semiconductor manufacturing, and fusion reactors.</p>