It would seem as though this is incorrect, since we each only have one life. Is my intuition correct that it should be everyone else's life and not everyone else's lives?
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Everyone else's life would be correct. In this case everyone is a singular pronoun. (See this explanation of everyone being singular.) Else is just a determiner used after everyone.
Allow me a digression that I think you might find helpful when thinking about whether to use life or lives in similar sorts of construction. The most memorable explanation of whether to use a plural or singular word that I've ever encountered comes from the book Words Into Type (p. 357). Under the heading and subheadings of Nouns, Number, Singular with a plural possessive there is this clear explanation: