The calligraphy flashback was Hime's grandfather (Kakushi's father-in-law, and the head of the family) declaring what her name was going to be. Kakushi and his wife didn't decide on Hime as her name, the guy who looks down on their marriage did, which is why Kakushi gets very reflective when Hime says it doesn't matter where a name comes from. Also I think you guys might have missed the photo they showed at one point in this episode, where toddler Hime is having a birthday party with both parents present.