When using MSA/formal speech, the tanwiin/nunation un/tun, which shows that the nouns and their adjectives are the subject of the sentence, are added. When speaking dialect or informally, the nunation is left off. Duolingo sometimes writes/pronounces sentences with nunation and other times without. The an/tan ending is used with nouns and adjectives that are objects in sentences and in/tin when they are possessives.
Yes, it's indefinite feminine subject (nominative). The nun drops if definite, I think. The / t/ letter we hear in the feminine nouns alerts us a word is grammatically feminine, which is often the case in Semitic languages in general. So whether it's tu(n), ta(n), or ti(n) ending depends on its function in the sentence (nominative = subject, accusative = object, genitive = possession). In other words, Arabic declines nouns to some extent based on how the noun functions in the sentence (similar to German). A curve ball is that these endings are not written in the sentence, but we hear them.
Correct. The case endings are only used when the words are part of a sentence, because the case endings indicate if the nouns and their adjectives are the subject of the sentence, the object of the sentence, or show possession. So, when you click on the individual words, the case endings will not be on the words. Also, sometimes Duolingo uses nunation in sentences and other times it does not.