Eric Garcetti, who plays a security officer for the mayor of Los Angeles, was the real-life mayor of Los Angeles at the time.
The pilot was written by Steve Carell and Nancy Carell (husband and wife, creators and executive producers of the show). Also directed by Steve Carell.
When Angie (Rashida Jones) takes off chase after the suspect, she quickly makeshifts a comically beautiful dress using nothing more than scrap drawing papers and a stapler. This bit is clearly referencing another show starring Rashida Jones, Parks and Rec. In season 5 episode 14, Rashida Jones' character, Ann Perkins haphazardly designs and creates a gorgeous wedding dress for her best friend Leslie Knope (Amy Pohler) using only scrap paper, receipts, and old photos.
The name of the claw machine the forensic team used to examine the ransom note is hown to be unguibus which is a distorted latin word meaning claw, hoof and a plural of unguis meaning finger.