The hologram machine that the Space Time Inspector gives to the Constable is a Hungry Hungry Hippos boardgame painted silver with the Hippos removed.
Annie's musical number, "Teach Me How to Understand Christmas," is a parody of a certain sub-genre of "sexy" Christmas songs, including "Santa Baby" and "Baby, It's Cold Outside." "Santa Baby" in particular is a target here, since its combination of sexual innuendo and the infantilizing implications of the lyrics have been criticized in recent decades as awkward.
During Troy and Abed's "Baby Boomer Santa" song, right as Abed begins his disco verse, the camera pans to the rest of the group, and Britta, Annie and Shirley can be seen doing the "hear no evil, see no evil, speak no evil", respectively, with Britta covering her ears, Annie covering her eyes, and Shirley covering her mouth.
During Troy and Abed's Christmas Rap, the "clock" which appears has "hour" markers with 10 images from past episodes. The 11th (highlighted) circle contains an image from the act and scene currently underway. The fifth circle is the on-board computer system S.A.N.D.E.R.S. from season two's Basic Rocket Science (2010).
The diorama figures Mr. Rad is using to plan the staging of the Christmas Pageant are the same figures made by the Study Group for their last Anthropology Diorama in Season 2, episode 21, Paradigms of Human Memory (2011), except Jeff's figure is missing.