I think it's a bit unfair of you to call this syntax "curious". As far as I know (and correct me if I'm wrong here), Modern English is the only language in the world which uses do-support (at least in this way), so if anything it would be curious if there were a word for "does" in this sentence. Of course it's understandable that this might seem unfamiliar to you, but it's the English language that has a "strange" way of constructing questions (and negations), not Danish.
I sort-of disagree with the point about English being the only language like that, but it functions in a similar way to "Est-ce que," in French and "Kas," in Estonian as they are both also interrogative particles. There are though many peculiarities of the English system such as negations as you mentions and in Indo-European languages switching the order of the word and noun is probably more common.
Which other languages do you know that use do-support in that way? "Est-ce que" certainly doesn't translate to "do", and it is also not necessary to construct a question, nor is it strictly speaking a particle (I don't know anything about Estonian, so I won't comment on that). And despite the do-support English still has the same inversion of word order when posing questions as other Indo-European languages like Danish or French etc.
How would it be "drink horse milk?", even if you were to say it directly it would be: "drinks a horse milk?" and that sounds ridiculous. If you understand the sentence (and I'd say if you directly translate it first then it's pretty easy) then you can also see that "drinks a horse milk?" sounds very unnatural.