This study focused on the influence of Mg codoping, non-stoichiometry and Ga-admixture on the scintillation and luminescence characteristics of Ce-doped lutetium-aluminum garnet crystals (LuAG), the stability of the Ce valence and the role of defects. For the Ce-doped LuAG samples, codoping of Mg at a concentration around 300 ppm led to improvement of overall scintillation efficiency, which is in accordance with previous studies and can be attributed to an alternative scintillation mechanism involving Ce4+ stabilized by Mg2+. Very significant improvement of the overall scintillation efficiency (almost a factor of 2) and exceptionally high light yield when compared to the stoichiometric sample was observed for the Ce0.2% Mg 300 ppm sample grown from the melt with 3 atomic % Al-rich nonstoichiometry. This was explained by enhanced energy transfer towards the Ce3+ luminescence center via defect levels induced by non-stoichiometry. The highest Mg concentration brings additional slow components to the scintillation decay. Very similar trends were observed for the Ga-admixed LuGAG, but the defect luminescence was strongly suppressed. Suppression of this competitive luminescence channel led to further improvement of the overall scintillation efficiency. Interestingly, X-ray diffraction has shown that the crystals contain two or even three garnet phases with very close compositions.