1)
It's "university/college" vs "department/school/faculty (of a university)".
e.g.
東京大学工学部
the School of Engineering of the University of Tokyo
2)
一度 means "one time", and it doesn't always refers to "a game", for instance, it might be just a practice/training as a not-so-common example. If you understand this, I think it's no problem to use "a game" for natural English.
3)
Yes. It's similar to やる気がある vs やる気が出る/やる気を持つ.
の is a nominalizer, so 日本に興味を持ったの literally means "Your taking an interest in Japan". As you can see, there is no connotation of "when" just by this phrase. It might be "why", "what" "where" etc., etc. depending on the context.
e.g
日本に興味を持ったのは、何がきっかけですか?(= what)
4)
そこ basically refers to "the location the speaker is talking about". In this case, it's the family who lives there, as you interpreted.
5)
Yes. で indicates the cause/reason.
e.g.
事故で電車が遅れた。
6)
Yes, は is the contrastive marker. It connotes, for instance, you might become to be able to read manga for children with no kanji/with okurigana, but you can't become to be able to read newspapers.
Because my studies are often, I intend to post often with things like this. I hope my own questions and answers are helpful to onlookers. If this is all outrageous to ask, I will not regularly post.
There is no problem at all. This is exactly the place for that purpose.