How is God essential to a dialogue, especially in Waiting for Godot or any kind of conversation?
The presence of God is essential to the discussion of Beckett's work in a few ways. The level of comparison that is the easiest to suggest is that "God" is part of "Godot," both literally and figuratively. The characters wait for something, a higher power, who possesses "the answers" and "will indicate what to do." In this respect, Monsieur Godot and God are quite similar, as the faith in the external force of Godot can be akin to the way one sees God. In Beckett's rendering, the figure of God is one who makes subjects wait, and the concept of waiting is one where individual action is met with paralysis. Individuals do not take action because they are "waiting" for this external force whose presence has not been felt nor experienced. This extreme level of hope for salvation or answers which arise from a totalizing force is seen in Godot and can be seen in the figure of God.
In Waiting for Godot, the absent Godot is a god figure. Estragon and Vladimir wait for him to appear, but in true existential fashion, Godot never arrives. This view of God is consistent with the existential idea that if God does exist, he is simply an obtuse presence who observes but does not involve himself in human affairs (The Great Gatsby's billboard is a similar portrayal of God).
In regards to the second part of your question, one could argue that what a person believes about God or a higher power--whether he exists or does not--is central to that person's world view. It determines how a human perceives his/her future, human nature, man's role in society, science, man's origin, etc.