Why can solid lubricants operate only at lower temperature in oxidizing environment but higher temperature in reducing environment?
Graphite is elemental carbon, and carbon is unstable in the presence of oxygen because they react to form carbon dioxide. The rate of this reaction: $\ce{C + O2 = CO2}$ is very slow and has a high activation energy, so a piece of graphite can sit in oxygen at room temperature for millions of years without anything happening to it. Once temperature increases, the reaction becomes much more rapid. From my own experience, a fist sized chunk of graphite can survive for several hours when heated to above 1000 °C in atmospheric oxygen.

Molybdenum sulfide is composed of two things that are likewise not stable in Earth's surface atmosphere: molybdenum and sulfur. It reacts by: $\ce{2MoS2 + 7O2 → 2MoO3 + 4SO2}$. As with graphite, this reaction is strongly temperature dependent.

There isn't a single temperature at which the lubricant is "not stable" any more. It is a balance of the required lifetime of the lubricant, the temperature, and how oxidising or reducing the environment is.