"The singer affirms that I sing better than her."
Translation:Người ca sĩ khẳng định là tôi hát hay hơn cô ấy.
I'm not very sure about this, but apparently "tốt" is mainly used for quality of a product, something you make physically. "Hay" tends to be used for idea or something abstract (like your voice or how you sing). But right now, I can think of "làm tốt hơn" (do something better) which doesn't follow that rule. You can also compliment an essay as "tốt" ("hay" is okay as well) but you can say that to a book or a poem.
i don't think that's right, because up to this point, "that" has been implied, and it is marked correct (usually) if you write it or don't write it (i have purposely tried it both ways). And "do" has been used. Now, all of a sudden, without explanation, it is saying "la," which goes against how "la" has been used previously (is/are/etc.), so there must be some new rule in use that they haven't explained for this sentence.
Well, you know, you're literally explaining to a native "that's not right".
I understand that you feel it's irrational when Duolingo didn't mention it before, but actually this happens quite often on Duolingo. They were just unaware of that, possibly. And that's why discussion feature is useful.