Doping in two-dimensional materials has emerged as an effective tool for modulating their electronic properties and thereby enabling their multifunctional applications. In this work, we present a first-principles study on induced effective magnetic moment and metallicity in MoS$_2$ monolayer by substitutional doping of group-5 transition metal (TM) elements -- V, Nb and Ta. From our study, we observe that the V doping induces half-metallicity, whereas metallic characteristics are observed in the case of Nb and Ta doping. Moreover, V and Ta-doped MoS$_2$ monolayers are observed to show total induced magnetic moments of 0.922 and 0.624 $\mu_{\rm B}$, respectively. Importantly, the combined effects of strong spin-orbit coupling (SOC), broken inversion symmetry, and structural asymmetry is observed to lead to a permanent valley polarization in the V- and Ta-MoS$_2$ systems. In particular, we observed a valley polarization of 121 and 21 meVs for V and Ta-doped MoS$_2$, respectively. Furthermore, an enhanced piezoelectric coefficient for the doped systems is observed compared to pristine MoS$_2$. Notably, the simultaneous presence of half-metallicity, substantial valley polarization, and enhanced piezoelectricity in V-doped MoS$_2$ establishes this system as a promising multifunctional platform for next-generation spintronic, valleytronic, and piezoelectric nanodevices. Overall, our findings provide fundamental insights into engineering coupled spin-valley-mechanical degrees of freedom in two-dimensional materials for advanced quantum and nanoelectronic applications.