Translation:My village is close to the city of Beirut.
In this sentence, the word مَدِيْنَة is considered grammatically to be in "construct state" with the following word بَيْروت, and thus does not have a definite article -- I think this may be briefly explained somewhere in Duo's "Hints & Tips", or one could check a reference such as the following (although that one is pretty technical, there are probably also references on the web at a more basic level) https://en.wikipedia.org/wiki/I%E1%B8%8D%C4%81fah --
Yes, we can! قريتي قريبةٌ مِن بيروتَ
Duolingo seems he/she wants to teach us about the structure of possession (2al-2iDaafa الإضافة) in this lesson. مِن مدينةِ بيروتَ is an 2al-2iDaafa form.
They are not. But that's not the question. The language that you and me are learning here belongs to the whole Arab world. We need to be presented with different aspects that can represent all of them.
I know that the authors are using the Arabic flavor from their own part of the region, but using proper nouns such as Dubai, Qatar, and Riyadh, or Emirates and Arabia wouldn't do any harm to the grammar, would it? I'm not taking sides because I'm from none of them but I hope it's not a discrimination arising from a feeling of resentment.
I had started my above comment with a question (Is there a reason...?) so that I wouldn't include my own guess, hoping that someday one of the authors would take the time to explain the reason here.