Translation:This is the key to my big brother's house.
Hrm. "This is my brother's house's key" isn't accepted, but "This is my brother's house key" is.
The Japanese sentence implicitly says it's the key of the house of your brother. The accepted translation only means it could be the house key of your brother for a house that is not his.
Eh, i think the way english works, house key is the preferred usage. Someone's house's key feels more unnatural than someone's house key. Because in this case house key, house is modifying the key to show what kind of key it is. Also, there's no ambiguity about whose house key it is, maybe just whether or not someone is actually living in that house, but thats irrelevant
かぎ 鍵 according to Jisho. But it also says カギ in katakana is OK. Not sure why exactly but in Satori Reader it has this to say. Sometimes writers choose to use kana even when there is a very well-known kanji. The word わたし is frequently written in kanji (私), but it's certainly not uncommon to see it expressed in kana. Again, this comes down to the writer's personal preference and the context they're writing in.
The author may choose katakana in order to draw attention to a word, similarly to how we use italics in English, or to indicate that they know the word has kanji but are deliberately choosing not to use it because they think it's too rare. For example, the word かぎ has the kanji 鍵, but it's often jotted down as カギ as a sort of shorthand for "You know the kanji that I mean." https://www.satorireader.com/help