When Lucifer studies the business card he says, "'Vatican Investigations.' It sounds like a soon to be canceled TV show," a joke to Lucifer being canceled after the third season and being picked up by Netflix.
Possibly for legal reasons, the poster for the Chewbacca sound-alike contest has a picture of a dog rather than a Wookiee.
This is the second time that Lucifer is seen using his ability to start machinery by force of will.
Amenadiel and Lucifer seem to think angel/human hybridization is unlikely. The Old Testament (Tanak) mentions "Nephilim," the offspring of "sons of heaven" and "the daughters of men." Jewish tradition does not interpret this as involving angels, but Christian tradition clearly calls Nephilim angel/human hybrids. Christian lore also says Heaven abhors the Nephilim.
The house that Chloe and Lucifer pull up in front of to confront the murderer is the house that was owned by Darren and Samantha Stephens of Bewitched (1964).