You're much closer to the actual Spanish pronunciation (especially if you mean a short e in 'rez'). The English pronunciation basically depends on whether you treat it like an English word with that spelling, or whether you do the best approximation of the native pronunciation with English sounds (like "Iran" as /aɪˈɹæn/ or /ɪˈɹɑːn/).
I agree that what the quiz shows is not how you are supposed to pronounce Buenos Aires. And it is a good question - whether to transcribe some of the names not as English text but as text in another language. Using Spanish generates ˈbwenos ˈajɾes, which is what JonOfKent's says.
Still, the quiz is made to be taken by English speakers and from the perspective of anglophone pronunciation, however much that distorts the native sound of the name. This makes what could be very obvious a bit more challenging, as in the example of bɛrˈliːn
Beijing is incorrect. The second syllable is a true J sound and not a ZH sound. The phonetic spelling should be like the initial J sound in Germany. I speak Mandarin and I am also an English as a Second Language teacher, so I know the phonetic symbols well. Nice quiz! I got 100%.
Still, the quiz is made to be taken by English speakers and from the perspective of anglophone pronunciation, however much that distorts the native sound of the name. This makes what could be very obvious a bit more challenging, as in the example of bɛrˈliːn