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've never heard "dome" used that way but PIV implies that "dome, domen" can be used very similarly to "hejme, hejmen", and has some citations from Zamenhof for "dome(n)".
So - you can use it, but it's probably better to use "hejme" as that seems a lot more common to me.
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.