a) So...this ~てしまう is used for showing surprise (that the the AC can make cool even in the summer). I'm not sure what you mean by "it's ironically used". Do you mean that she's not actually surprised that the AC can do this?
Also, I found this sentence. Here, is it "to complete"?
屋敷には200個のオレンジを30分で絞ってしまう機械があった。
c) Ahh, then 行ってしまう just means "to finish leaving" (what is being completed is the departure part), right?