"МойбратживётвМоскве."

Prepositional case - Москве is the object of the preposition В.

Москва is Nominative feminine singular.

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.

In Spanish it's Moscú.

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.

Try and listen to it on Forvo.

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.

It's just third person singular.

"мы" is a personal pronoun, "мои" is a possessive pronoun

I took it to mean Locative Case, PalaeoJoe.

Locative case (also known as Prepositional case)

location