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?
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: