In English, "does" would only be used here for emphasis if it were a rhetorical question. "All this paperwork we have to fill out seems to assume we each have several secretaries at our disposal - but who does have several secretaries?"
So it might be okay in that one special case, but in general it would be a poor translation, so it should not be accepted.
"Have" would only be correct if the "who" referred to a group of people (and to the people, not the group, ie. "the senior managers", not "the senior management team"). I'm struggling to think of an example. Perhaps if you heard someone saying, "They have several secretaries working for them," you would understand that the "they" was some group of people, and you could ask, "who have several secretaries?" Even in this context, where it's probably correct, it sounds quite odd though.
So first of all: 'vem har' can both mean 'who (singular) has' and 'who (plural) have', right? This is how I interpreted this question: suppose you have a group of senior managers. Several of them could have more secretaries. So when you ask 'who have several secretaries?' you're asking which managers have more secretaries. That's not so odd, is it? I must admit English is not my native language, so please correct me if I'm wrong on this one.
Oh, I just realised something else about this, or at least I think I have:
As I understand it, "fler" means more; "flera" means several. So if it were the question, "who has more secretaries [than the others]?", I think the Swedish question would be "vem har fler sekreterare?" A very subtle difference!
According to this, only flera can mean several, but either can mean more, which sounds right to me:
Swedish is not my native language (English is), but as far as I know, the given Swedish phrase does not give a clue as to whether the 'vem' is singular or plural. So in that sense it might be that you're correct.
However, a native English speaker would not usually use a phrase like "Who have several secretaries?" -- it just sounds strange -- so it would be a bad idea for the course to accept that as an answer.
In your example we might ask something like, "which of the managers have several secretaries?" -- and even then, only if we already had an understanding that more than one of the managers does have several secretaries.
However, it sounds so unnatural that I think most native speakers would choose "has" over "have" in this context, even when the subject is plural. (I don't have a formal understanding of English grammar, but I would not be surprised if the rule was that "has" should always be used in this case.)
The only place this phrase would sound natural to me would be in a subordinate clause: "The managers, who have several secretaries, need not lift a finger all day."
who as a relative pronoun can have either singular or plural (in your example with the managers).
As an interrogative pronoun, it can have the verb in the plural if it is used to ask about the rest of the sentence, like in Who are they?, but otherwise the verb is in the singular even though it may in reality refer to many people.
In Swedish, if you know for a fact that you are asking about several people, you can say vilka, like in Vilka har varit ministrar?, literally 'Which have been ministers?' (= 'Who has been a minister?') You can stress the possibility of several people by asking Vem eller vilka har ätit upp kakan? ('Who or which ones have eaten all of the cake?') but you can also just ask Vem …? without implying that you definitely refer to only one person.
Not quite the meaning you're going for, but these two examples came to mind:
- "Who have you killed?" — ie, "Who did you murder?"
- "Who has you killed?" — ie, "Who was it who has put the order out for you to be killed?" or, more likely: "Who the hell would go to the bother of having you knocked off?!"