This phrase has no auxiliary verb, and therefore should not be in the ( strengthening) section on auxiliary verbs
Duolingo often throws in a sentence irrelevant to the topic of the lesson. I've seen this numerous times in this and other courses too.
No, it is not -- you need the definite article in Greek as well in this construction.
Similarly with e.g. το πεπόνι μου for "my melon"; you cannot leave out the definite article there, either.