could the story exist without the class differences between Amir and Hassan?

I don't think this would be the case within the context of Afghanistan. Amir and Hassan are so intricately linked through a rigid culture system which dictates the whole story. Amir is a higher class than Hassan and Hassan is of course a dreaded Hazara or "half breed". This limits their relationship to certain parameters. Amir controls the kite while Hassan is the runner. This is a symbolic distinction that permeates their relationship. When Hassan is raped by the village bully, Amir has no legal or culturally moral reason to intervene. He doesn't intervene and the very human guilt haunts him forever. Amir's guilt even seems to transcend Hassan's humiliation. As the story progresses, class distinctions become blurred. In the end Amir adopts Hassan's child and moves to the United States. This even further erodes the class distinction that limited their relationship at the beginning of the novel.