It's because it's a question. In Spanish, most questions, especially those that start with the English "Wh-" words (like where, who, what), have reversed order for subject and verb. That is, the verb should come right after the question word and the subject comes next.
We almost do this in English, except we put a "do/does" or similar word right after the question word instead. Like "Where does your sister live?" or "What does Charlie want?"
I'm pretty sure Bruno_NotD is pulling your leg. There is virtually no difference between American and British English grammar when it comes to "do" as an auxiliary. Though there certainly may be differences in usage of phrases such as "have you any" vs. "do you have any," that usage is irrelevant to the sentence in this Duo drill.
It is just my opinion, but I'm confident that Queen Elizabeth would prefer "where does your sister live?" over "where lives your sister?".
It is vive because the subject is tu hermana (your sister).It would also be vive if the subject were tu hermano (your brother) who is a "he". The gender does not change the verb. The difference between these is if the subject is a "you" (vives) or "he/she/it" (vive).
It would be vives if the subject were tú (you) note the accent over the "u".
This is why it's important to pay attention to the accents. "Tú" (with an accent) is "you" which is a subject pronoun. "Tu" (without an accent) is "your" which is a possessive pronoun.
Technically not true Daniel, although the forms without the helping verb are bound to sound old-fashioned and "strange" to modern speakers.
As an example, consider the nursery rhyme: "Baa baa black sheep, have you any wool?"
Will we say this is incorrect, and force children to sing "Do you have any wool?" :)