Logic would suggest that a dog being eaten by a piece of paper is a highly unlikely occurrence. But, if it magically did happen -- this is Duolingo, after all -- we would have tvého psa (accusative) rather than tvůj pes (nominative) in this sentence. This is an example of the flexible word order in Czech, which trips up so many of us, so often. :-)
But the word order that you suggested is a valid sentence, meaning "Your dog ate a/the letter."