Let's break down the sentences:
* "When the storm began to abate we went home." — Correct.
* "By light of moon it tripped over rock." — Dangling modifier. It's unclear what "it" refers to.
* "The dog that is mans best friend cats worst enemy?" — Grammatically incorrect and confusing.
Correction: "By the light of the moon, I tripped over a rock."
Correction: "Dogs are often called man's best friend, but they can be a cat's worst enemy."