Dumbledore left Harry at Privet Drive because of the protection against Voldemort that his blood relatives would offer him
Does this mean Harry is protected as long as he is with Aunt Petunia, or as long as he specifically lives in Privet Drive? If the Dursleys moved house, would Harry have to move with them too, or could he live alone in Privet Drive with the Dursley's permission?
The protection specifically applies to the house where Petunia lives. If Petunia moves house then so does the protection.
“While you can still call home the place where your mother’s blood dwells, there you cannot be touched or harmed by Voldemort. He shed her blood, but it lives on in you and her sister. Her blood became your refuge. You need return there only once a year, but as long as you can still call it home, there he cannot hurt you."
(Order of the Phoenix, Chapter 37, The Lost Prophecy).
Obviously, the protection applies to the house and not to Petunia herself; Harry doesn't have to be physically nearby Petunia to get the protection. She can leave the house for the evening and Harry is still protected. What is important is that Petunia "dwells" there, that both she and Harry can call that place home.
I'm not going home... not really?
The spell from which Harry had protection was due to, as stated, the spilling of his mother's blood. Petunia technically has some of the same genes as Lily, but obviously we can share parents and be opposites in all ways except for who are family is.
Harry has to return to a dwelling with his families blood once a year. This keeps up the charm. He could have lived with the Potters but Harry's father and his family were all dead and James was the only son of his Parents, thus Petunia was literally the only choice. The last blood relative.
It had to be, if I'm not mistaken, a direct relative, not distant such as a cousin.
P.S. This is only while he is underage. So he gets his protection from his aunt's blood attached to the charm set in place.
