Indeed, Arwen does decide to come back to Aragorn just a few days later. She marries him, thereby becoming mortal. The last of the Fellowship who were staying with King Elessar and Queen Arwen ...

Aragorn is called Strider due to his Ranger ... At various points throughout the story, he's also called Isildur's Heir, Estel, and Elessar, as different characters know him for separate reasons ...