It is related. Someone else can probably give a more grammatically rigorous explanation, but "можете" is "(you formal) may" and "мо́жешь" is "(you informal) may" (in the sense of may = allowed to). Then можно is a form of this word which is used in a different case (sorry! I'm terrible at remembering them, it's been too long since I learned them properly). You could say "можно мне здесь жить?" or "Вам/тебе/нам/мне/иму можно здесь жить." But to use the other forms/conjugations you mentioned, the sentence would be more along the lines of "Вы можете здесь жить" or "Ты можешь здесь жить". Different pronounce, different case.
I realise this is not the best explanation ever, but I hope it helped at least a little. :-/
Since I/we isn't used in the Russian sentence, which is asking about the general habitability of the place, a direct translation should preserve the ambiguity. Without further context the sentence could range in meaning from "Is it possible to live here?" to asking whether you can stay or move in.
I don't understand why "May I live here?" is marked incorrect.
In other examples, «Где можно курить?» was allowed to be translated as "Where can I smoke?" What's the difference? It's almost like there's some hidden context that we're supposed to assume without ever having been told.
What if a child was asking about a national park, would that not be one valid context -I am sure there are plenty we can think of; the point is to use an appropriate example to illustrate a valid grammatical point or to enhance our vocabulary in order to learn the language without any of the former being mutually exclusive: that is not always easy in short sentences in the beginning, especially, it seems, between Russian and English, yes?!
Is the audio correct for the word 'жить' here? It sounds like "Zhreet' to me and I was under the assumption that it should sound like 'Zheet'. I know it's a small difference but it threw me off for a little bit and I thought I had missed a new verb somewhere.
I checked forvo and it seems like my interpretation may be favored, but I'm not entirely sure now. https://forvo.com/word/жить/
Depending on the level of abstraction, "Can we live here?" or "Can people live here?" would be more natural in English. While correct, your translation would never be used in modern spoken English and you would only find something like this in books hundreds of years old.