Harper's badge is the standard Police Officer embossed badge. As a Detective her LAPD badge should be embossed with "Detective" (she wears the correct rank insignia for a Detective).
Madeleine Coghlan (Abigail) was in the Castle episode "PhDead (2015)". Madeleine's mother, Katherine Kamhi also appeared in the episodes, Target (2013) and Hunt (2013).