The use of “whoever” in these two sentences you presented isn’t permissible at all:
(1) “Whoever’s this is is to be returned.”
(2) “The office is cleaned by whoever’s turn it is that day.”
The pronoun “whoever,” which means “whatever person—no matter who,” can be used in any grammatical relation except that of a possessive; that is the prescribed rule in English. This means Sentence #2 above is an inherently grammatically flawed construction, for the phrase “whoever’s turn it is that day” uses the disallowed possessive form “whoever’s.” Another serious flaw in Sentence #2 is that it makes the noun “turn” in “whoever’s turn it is that day” the doer of the action of the verb “cleaned.” This is not possible in reality, of course; in the context of that sentence, only a person can do that cleaning job. And even if we acknowledge Sentence #2 as possible colloquial usage, it needs to be recast to the following form to be grammatically acceptable: “The office is cleaned by whoever has the turn for that day.”
The use of “whoever’s” in Sentence #1 is likewise grammatically flawed for using the disallowed possessive form “whoever’s.” Worse, the use of two successive “is’s” by that sentence is syntactically very clumsy. Here’s a possible rewrite that can salvage that sentence into acceptable form: “Whoever found this (“bag,” “purse,” “iPod,” whatever) has to return it to its owner.”
