In general, GSH is important for a number of metabolic events. It has an important role in - ascorbate-glutathione cycle (wherein it can help in conversion of dehydroascorbate into ascorbate, while getting its self converted to GS-SG), - synnthesis of phytochelatins/metalothioneins, -can sequester several heavy metal ions, - can scavenge ROS.
There could be a possibility of higher/increased activities of CAT, SOD, GPx, GPOx as well as levels of GSH in living systems or their components under stress. Therefore, the possibility of an increase in activities of these antioxidant enzymes due to increase in levels of GSH is little hard to accept.
It depends on the type of stress. Some heavy metals induce phytochelatins synthesis, and GSH is a precursor of phytochelatins. According to the stress level, antioxidant enzymes could rise their levels and GSH decrease it because it is used for phytochelatins synthesis. Everything can happen regarding the antioxidant response
I think that if a cel is subjected to an oxidative insult, usually it responds to face uo the stress deriving from it. In my opinion your system is responding in the right way, like an engine running at full. It is plausible that if you controls the expression of GSH-synthesizing enzymes (mostly GCL) or other proteins related to GSH homeostasis (e.g. gamma glutamyl transpetidase) they should be increased as well. Otherwise, what is GPx increase for, If it misses its cofactor? I believe that you are observing an oxidative/electophilic stress response activated by Nrf2. Try to analyse the expression of genes activated downstream of it (e,g, Nqo1).
There is no clear relationship between GSH levels and the realm of 'antioxidant' enzymes. GSH biosynthesis is under the control of the NRF2 system, as are some of the peroxidases, not all. For sure, the 8 GPxs, the 6 peroxiredoxins and the 3 SODs are not regulated the same way. Read the relevant literature before starting meaningless measurements (small collection attached).
My experience over several studies that the levels and activities of antioxidant enzymes are not well correlated with the levels of GSH or free radicals generated