The key here is Heavy Noun Phrase Shift. Here's how it works:
First, let's start with the canonical (declarative, affirmative) version:
[The person who acts upon this advertisement and accepts the offer]subject puts himself to some inconvenience at the request of the defendants.
Then we'll add the dummy auxiliary do. We need this auxiliary for two reasons: to negate the predicate, and to mark the clause as interrogative through inversion:
[The person who acts upon this advertisement and accepts the offer]subject doesauxiliary put himself to some inconvenience at the request of the defendants.
At this point, we can negate the predicate:
[The person who acts upon this advertisement and accepts the offer]subject doesauxiliary notnegator put himself to some inconvenience at the request of the defendants.
And now we can invert the subject and auxiliary, turning our declarative clause into an interrogative clause:
Doesauxiliary [the person who acts upon this advertisement and accepts the offer]subject notnegator put himself to some inconvenience at the request of the defendants?
Last, because our subject is so long and syntactical complex, and because this is particularly formal English, we can optionally make use of the rare Heavy Noun Phrase Shift, pushing our subject to the right out of its basic position, past the negator:
Doesauxiliary notnegator [the person who acts upon this advertisement and accepts the offer]subject put himself to some inconvenience at the request of the defendants?
When these conditions aren't met, we can't shift the subject noun phrase to the right:
Doesauxiliary shesubject notnegator like ice cream?
*Doesauxiliary notnegator shesubject like ice cream? (ungrammatical)
The subject she isn't long or syntactically complex enough to count as as a "heavy" noun phrase, so it can't be shifted past the negator.
In contrast, the contracted form doesn't doesn't involve Heavy NP Shift; -n't is a suffix, and so doesn't is a single word which inverts with a subject without any of these restrictions:
Shesubject doesn'tnegated-auxiliary like ice cream.
Doesn'tnegated-auxiliary shesubject like ice cream?
For more discussion, see my previous answer.