Answer: Negi's father is the leader of the family.
The most likely situation is that Negi's father is the leader of the family. This is likely because of his actions. If Negi's father is able to intervene in a way that means she does not have to help her mother, then it is likely that Negi's father is the leader of the family, and that the other members tend to do whatever he believes is right.
Simply, the element repeated is the redundancy of the characters' name. Papa says to stay inside the gate. Papa goes away for something. Little Daughter looks through the gate.
Personally, i would have stayed. I mean, cmon! He would have stayed safe and lived an eternity just having fun and playing games! But I get why je left. He felt it was his duty to complete his mission. Also, he wanted his mother back, and dint want his freinds to die