熱い空気 と 暑い部屋 はどう違いますか？説明が難しい場合は、例文を教えて下さい。

The translator is right.
I guess 暑い is fit for the hot environment rather than climate, and 熱い for career material of heat.
When you specify the environment by pointing to the place or the time, it is 暑い. So a hot ROOM is 暑い. And when you say "I feel hot", say 暑い, it's maybe because it means I am in a hot environment so I feel hot, but not that I am hot in a material term.
On the other hand, 空気 is a career of heat, so 熱い. In awfully hot day, you would get annoyed by a hot wind blowing onto your face, this is 熱い空気 or 熱い風. If you say 熱い部屋, it sounds weird but I guess you would mean the walls of the room are hot.

暑い夜、熱い水、熱いカップ are right, except for metaphor.