Is there a reason why "The dog is under the window." won't work? Is there anything in this sentence that tells us that this is an unspecified dog, rather than a specific one?
Yes. The word order. Please see my comment on a similar sentence: https://www.duolingo.com/comment/11607192
"Under the window, is a dog" makes perfect sense and doesn't add "there" where it isn't in the Russian
Couldn't this whole phrase be its own subject, denoting a dog that happens to be under a window (and thus translated as "the dog under the window")?
The natural word order for a nominal phrase would be «собака под окном».
That would be «Собака под окном».
‘The’ suggests the dog is known to the listener, and we put known things towards the beginning of the Russian sentence.