It is not about "does" affecting a verb. It is about what the verb is.
The main clause is the question "Why does she ask [...]" The verb phrase here is "does ask", and it is in the third person (does) because the subject is "she"
There are two more dependent clauses. An infinitive clause "to stand" and a dependent clause "every time she enters the room". The verb in this separate clause is "enters" and it is also third person singular. As this clause isn't a question it doesn't have an auxiliary verb.
The form of the verb in each clause is dependent on the subject of that clause, and not on the structure of any other clauses
Why do you ask him to stand every time she enters the room?
Why does she ask him to stand every time I enter the room?
Why does she ask him to stand when she enters the room?