Multi-valent (MV) ion intercalation batteries that replace the Li⁺ with a MV cation such as Mg²⁺ provide a promising approach to meet the high energy density demanded by the next generation of electrical devices. One of the challenges in achieving high energy density MV-ion systems is to develop a suitable cathode with a high enough voltage and diffusivity of the MV cation. Mg intercalation into Orthorhombic and Xerogel-V₂O₅ is one of the very few that has been shown to function reversibly at reasonable efficiency. In this study, we gain insight into the thermodynamics of Mg insertion into various Orthorhombic V₂O₅-polymorphs (including α and δ) and the Xerogel, from first-principles calculations. While we have calculated the 0 K phase diagram and the equilibrium voltage curves for all polymorphs, we have computed the migration barriers for Mg diffusion in the α and δ-V₂O₅ polymorphs and find significant influence on the Mg mobilities by the coordination environment in the respective polymorphs. We have also evaluated the performance of α and δ-V₂O₅ for other MV ions including Ca²⁺, Zn²⁺ and Al³⁺. Furthermore, we evaluated the role of H₂O in the intercalation of Mg in the hydrated Xerogel-V₂O₅ structure and found that H₂O+Mg co-intercalation can happen based on electrolytic conditions with consequent impact on the voltage and phase behavior of the Mg-Xerogel system. We believe that this study can be further used to improve the performance of V₂O₅ as a cathode material for Mg (and MV)-batteries. This work has been done as part of the Joint Center for Energy Storage Research (JCESR).