club is the short form of "nightclub" - so both should be accepted, as should "When does the club/nightclub open"
I would argue that similar to English, you can say... "When does the club open?" OR "What time does the club open?". While these mean the same thing they are not the same words.
Same here, where you can say this probably 10 different ways, but as it stands "a que horas" means "at what time". "Quando" means "when".
Club has lots of other meanings, so "nightclub" would depend on having the context. Your translation of the whole sentence is correct, and more natural in English than the awkward one given.
IMHO this kind of questions should be more difficult. It's almost impossible to give a wrong answer when sentences are so different.
Is this meant as as an astonished remark more so than an a genuine question? If it's meant as genuine question, surely "what time does the club open at" should be an acceptable solution, given that's how we would most commonly order the words of the question in English.