The "n" indicates indefiniteness. The vowel before it indicates case. "u" for the subject, "a" for objects, and "i" to show posession or relation, as in "kalb-un bint-in 'akala." = "a girl's dog ate.
Nouns and the adjectives that describe them should always have the same ending, even though the computer is getting it wrong and sometimes pronouncing some of them as in classical arabic, with the full declension, and some of them as in modern dialects, without the "tun".
You can't really mix and match like this in real life.
Yes. In classical arabic, a noun which is a indefinite subject is declined with ُُـُُُُُُـُ at the end, even though it is not apparent in the sentence here. However, nobody declines the words when it comes to speaking arabic. It will be pronounced "bint 3arabiyya dhakiyya". In fact the audio says the sentence as if it was something read out loud from a book, because Duolinguo teaches arabic which tends more to literal arabic rather than spoken arabic. Sorry if my explanation is confusing, maybe a native arabic speaker could explain this more clearly.
Exactly. When the ة get on of the Tanweens "un-an-in" the ة get the T sound like the letter ت + the tanween.
By the way , not all words ending with ة means they are feminine. The name Hamza حمزة is an arab masculine name. Also the adjective red ( for feminine things ) is حمراء not حمراءة red ( for masculine things ) being أحمر. However you won't encounter alot of these words so don't worry at all they are the exceptions.