3. The Japanese girl is, I think, talking about some of the things she criticised the American girl for.
一つは、シャワーの使用を許したこと(冬になってもシャワーしか使わず、風邪をよくひいた)。
I assume the start of this means "For one thing, there was the matter of the permission to use the shower..." (i.e. this was one of the things she nagged about).
Then there is something about using the shower in the winter and catching colds, but I don't understand the overall sense or who is the subject of the verbs.