Q. Is “this is mine and Kelly’s cat” correct? Would you please explain the rules behind this sentence. Thank you.

A. Sorry—not even close. Think about it: would we say, “This is mine cat” (if we are not Shakespeare)? Rather, we say, “This is my cat,” because the word that precedes a noun (cat) as a modifier must be an adjective, like my. (The pronoun mine, however, can serve as an adjective when it’s placed after a noun: “The cat is mine.”) And since it’s polite and grammatical to put yourself last in a list of people (a custom that dies a little more every time someone says “Me and Jughead bought us a pit bull”), the correct phrasing would be “This is Kelly’s and my cat,” or, less awkwardly, “This cat is Kelly’s and mine.”