"The eagle eats a mouse."
Translation:Ο αετός τρώει ένα ποντίκι.
Translating it back to Greek shouldn't it work even without indefinite article?
Not really, the meaning changes a bit. Without the article it would be more like "The eagle eats mice", as if a mouse is part of the eagle's diet, the same way someone may ask "Τρως αρνί;" = "Do you eat lamb?" Answering yes does not mean that you're eating a lamb, but you will eat it as food. The lesson sentence is more specific in its meaning than that.