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EuTiO₃ has drawn interest in recent years, as it exhibits paraelectricity similar to SrTiO₃ and KTaO₃. Moreover, this insulating perovskite is a candidate quantum multiferroic, as it exhibits antiferromagnetic (AFM) order below T_N ~ 5.5 K; potential ferroelectric and magnetic quantum critical points might be reached by alloying and/or strain-tunning. Furthermore, prior theoretical and experimental studies of metallic EuTiO₃ have shown that carrier-doping can tune the ground-state spin configuration from AFM to ferromagnetic (FM). Considering the possible connection between ferroelectricity and superconductivity proposed for SrTiO₃, a search for superconductivity in metallic EuTiO₃ is of considerable interest. Here, we present a novel annealing technique to introduce oxygen vacancies and, hence, metallicity in floating-zone-grown insulating EuTiO₃. We leverage charge transport, magnetization, and x-ray diffuse scattering measurements to investigate the electronic properties of doped EuTiO₃, specifically: the cross-over between AFM and FM ground states, searches for superconductivity and multiferroic quantum criticality.