Prepositional case - Москве is the object of the preposition В.
Москва is Nominative feminine singular.
Is this verb used for living in general or just for living in some place (like German wohnen)?
"живёть" means both. as far as I know Russian does not distinguish between those two meanings (similar to "kennen", "können" and "wissen").
How did the pronunciation "Moscow" come about? Why doesn't English say the city by it's original name?
I wish I had a good reason for you ;) It's the same reason we don't call Spain Espana, and don't call Italy Italia, etc. etc. etc.
That would probably be an interesting etymological (in English) and pronunciation investigation. I see two possibilities, to begin with: the British with their inimical way of mispronouncing foreign words bequeathed us "Moscow". Another is the Russian connection through Alaska, and possible filtering through Inuit, British, and French speakers in Canada and the Yukon, gradually making it to the lower 48 States of the US.
As a note to this: the letter sequence "kva" is not something you encounter in English. People hear what they are used to hearing rather than what is actually said, so there's a good chance people just didn't hear it right, and said it as best they could, using standard English. Another is that nobody heard it pronounced correctly at all, and someone who didn't know Russian made an effort to transliterate the Russian into something approaching English.
Those are all guesses, though. It's an interesting topic.
That second possibility does not make any sense. The word for Moscow is very similar on Dutch and German.
Moscow has been around for a very long time. And the pronunciation of its name has evolved. And the English have known about it for much of the time. Both Russian and English languages have evolved. And the pronunciation changed in different ways in the different languages. In English we used to talk about "the Duchy of Muscovy" - which is much closer than the modern English form.
It's a preposition, meaning "in/at" and probably a few other things. москве is in prepositional case, because it's the object of preposition В.
If going just by listening, the sentence sounds like a question which I think is a mistake
When I mouse over "Москве" the drop down has "(Loc./Dat.)" at the bottom, the second part is "Dative Case" but what does "Loc." represent?
To me it sounds like she is asking: does my brother live in Moscow? She clearly raises the voice at the stressed syllabus