Greek myth does not say, but given that she was daughter of Demeter and Zeus it is likely that Persephone was born upon Olympus itself.
Persephone did not have a favorite: as all people die whatever their city.
Persephone does not have a personal city: the people of all cites die and come to the Underworld where she is Queen.
All people die no matter the city, so no Persephone was not a patron goddess of any place or city.
Persephone was born before time existed.
All people who die go to the Underworld, so Persephone need not have a favorite city.
Persephone was born in a meadow on Earth
Persephone was born to Demeter from Zeus.
All people die no matter the city, so no Persephone was not a patron goddess of any place or city.
Persephone was born to her father Zeus and her mother Demeter, on Mt Olympus.
Persephone was born of the goddess Demeter and the god Zeus, so was born with her "powers".
Yes, Persephone is associated with the city of Eleusis in Greek mythology, where she was abducted by Hades and later became Queen of the Underworld. The Eleusinian Mysteries, celebrated in Eleusis, were centered around Persephone's story and her role as a symbol of death and rebirth.
Greek myth does not say where Persephone was born.