Are all healthy animals more likely to defecate near the end of a rest-cycle? Just what the title states. It stems from observation & personal experience that a person/dog/cat/monkey is more likely to relieve oneself immediately after it wakes up from the peak-sleep cycle of it's body-clock. Is this observation true? What causes this behaviour ?

I think we can actually go farther than mere behavioral argumentation. The separation of the autonomous nervous system into parasympathetic and sympathetic is, one the one hand, associated with the sleep/wake cycle, on the other with parasympathetic/sympathetic activity (low/high epinephrine secretion). This duality certainly applies to all vertebrata (i.e. big animals), as fight & flight is associated with all vertebrata, see e.g.

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In other words: from the existence of fight or flight in all vertebrata I infer the existence of a dual autonomous nervous system (parasympathetic and sympathetic) in all vertebrata, although there may be other proof. Since in sleep the parasym. NS is active, the bowel moves. Conclusion: the bowel moves in the sleep of all vertebrata.