Где они живут
Some sentences should be written in cyrillic, so that the readers can understand the correct pronounciation.
They can all be in Cyrillic. There is a button on the top left of the lesson page that you can change the transliteration to Cyrillic.
In "живут" why "и" sounds like "a"?
I suppose, it's just a tiny error of the robotic voice =)
Yep. That's not how it's actually pronounced, и always sounds like "ee" as far as I know.
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What is the infinitive of живут? Why isn’t it живят?
The infitive form is жить, according to the dictionary.
The lesson explained that они́ needed -ют (ут) / -ят (ат) at the end of the verb, but we don't really know which one we have to choose : why isn’t it живют? How can we know ?
The conjugated verb for они is ют isn't it? Not sure why it becomes живут...