The audio gives "he'ayef" rather than "ha'ayef" - is that a common sound shift ?
It's not used by Modern Hebrew speakers. See here - https://www.duolingo.com/comment/16826077
The speaker uses here the traditional form of the article before unstressed עָ, i.e. הֶעָיֵף
No, it should be translated, the dog is tired, he doesn't want to eat.
The dog is tired = הכלב עייף
The tired dog = הכלב העייף