How did Egyptian theology dominate the lives of all caste members?

First of all, the Caste System belongs to India; ancient Egypt never had it. Egyption religion held that everything that happened in nature was divinely inspired, from the rising and setting of the sun every day to the growing of crops or even to the behaviour and existence of rats. So they had a great number of gods, from the Sun god to the god of rats. That meant in order for nature to function normally, all those gods had to be kept pacified which in turn led to the existence of a very large priestly class. They were the ones who did most of the religious work; it did not really 'dominate' the daily lives of everyday citizens. But we do know that brief incantations were often said before any action that might cause or necessitate any action of the god involved.