The word جارٌ is introduced early in A. S. Tritton's Arabic (Teach Yourself Books; chapter 3, p. 37), perhaps, as you indicate, because it's a simple word, but also the word can refer to a "protected outsider" and so it's an important word historically for the ethical treatment of strangers in an honor-shame culture.
Some teachers provide first words that may be uncommon, but - more importantly - are short and follow common patterns (eg In English, I was taught words like cat, bat, hat at the beginning, and then brat, that,...). I'm hoping that Duolingo is following this approach.
Also, more abstract words like pronouns/ verbs are more common, but personally I remember the simple, concrete nouns (eg house) more easily.
No. Not always. It is called tanwīn in Arabic. It is Arabic equivalent of an indefinite article. It is just the consonant 'n' sound connected to the previous letter.
Kitaabu-n : a book Al-kitaabu : the book
Plus it is generally written in Arabic as a diacritical mark
كِتَابٌ - read كِتَابُنْ (not كِتَابُون)
Hope it helps.
Try not to let it distract you. It frustated me a lot when I started learning Russian on Duolingo, but I have discovered that Duolingo's method works. It helps to change your perspective on "mistakes" - I took a long time to realise that I'm not getting minus points for not knowing something I wasn't told. It's alright to guess in the exercise and then you'll be told the meaning. Next time, you'll probably know :) It's more of a learning process, than a one-off test.