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."
I think that's not a purpose of Duolingo to describe realistic situations.
That said, the sentence makes perfect sense when someone sends you a letter, you ignore it, then meet the sender who asks about the letter, and you say this to make it obvious that you intentionally ignored it. There was a Southpark episode where Cartman explains to Kyle where did his invite card get lost.
I am native AmE, and I see nothing wrong with "A dog ate your letter." But "The dog ate your letter" may also be an accepted translation. If you used it and it was rejected -- did you report it, if so? -- either I'm wrong about that or there was something in your answer that Duo didn't like.
You're sitting at the dog park chatting with a dog-owning friend about a letter you just got from someone you'd really like to get to know better Your dogs are off having some fun.. Suddenly, you see that a couple of Bad Boy dogs are circling your dog in an unfriendly-looking way, so you toss the letter down and make a beeline for the dog mugging, The Bad Boy dogs see you coming and run off; your dog happily trots back with you to the bench, feeling safe in her owner's presence again.
But wait... your special letter is missing! In answer to your question about what happened to it, your friend says, "A dog ate your letter! He gobbled it up before I even realized what was happening!"