Looking at their entries in PIV, maŝino = machine, and aparato = apparatus. Maŝino is used for mechanical devices, and it usually refers to a motor or engine. Aparato is for all kinds of apparatuses (they don't have to mechanical), as well as systems like body systems (e.g. la digesta aparato = the digestive system) and organisational systems (e.g. la ŝtata administra aparato = the state administrative apparatus).
In this sentence, I guess they prefer aparato as the answer because the device mentioned is not an engine.
I would understand "dome" to mean "at the house" speaking about a house in particular, not necessarily someone's home.
Correlatives! I love the idea of them but I cannot hear the difference between so many of them. So frustrated.
I agree that in the present audio it is hard to distinguish between "kiun" and "kian." I typed "kiun", it was accepted but the discussion page shows "kian." But that seems to be a problem in the audio, which could be clearer.
Generally I find it more difficult to distinguish between "mi" and "ni" in DL's listening comprehension, though that is never a problem for me in conversation -- maybe because of context.