Anvilicious: Death's speech to Susan about how humans need fantasies like the Hogfather (i.e. Santa Claus), or Terry Pratchett's books for example, to believe in when they're young so that when they grow up, they can believe in big fantasies like justice, mercy and...purpose.
The point was that something doesn't have to be a concrete idea to be a valid one.
Complete Monster: Jonathan Teatime is a sadistic assassin feared even amongst hardened criminals. Implied to have killed his own parents, Teatime has a penchant for excessive slaughter. In one instance, Teatime nailed his target's pet dog to the ceiling for barking too much and killed his servants because he didn't want them to barge in on him. After the Auditors hire Teatime to kill the Hogfather, Teatime abducts a tooth fairy to use her magic to make all the children in Discworld stop believing in the Hogfather in order to kill him, causing a void in Ankh-Morpork's magic that wreaks havoc in the process. Teatime, unlike other members of the Assassins' Guild, shows no regard for the lives of civilians or his own men, in one instance killing a stagecoach driver out of boredom while chatting with him and in another instance having a locksmith working for him dropped down a long spiral staircase to a painful death when he's no longer useful. When Death and Susan foil his assassination attempt, Teatime tails them to the nursery Susan works at in an attempt to kill Death.
Ensemble Dark Horse: Teatime is quite the popular character among fans, despite only appearing in this one book.
Genius Bonus: Many of the computer references made towards Hex. In particular the teddy bear being removed might be a reference to the Microsoft Teddy Bear which was used as the icon for a vital file in the early windows OSs. Many users (helped by an email hoax) mistook the strange file for a virus and deleted it, with disastrous consequences.
Iron Woobie: Death. A recurring theme from his own Myth Arc, true, but it's especially visible when you see just how much he prefers giving presents to the living instead of guiding the dead.
Nightmare Retardant: In-universe. Gawain and Twyla don't find Death scary, thus not identifying him as a "monster", partially because he's on a big comfy chair and eating a biscuit.
Uncanny Valley: The kids find Teatime's freaky heterochromia is a great deal scarier than a talking skeleton. Particularly because, as mentioned above, the skeleton is sitting in a comfy chair and eating a biscuit, and because Mr. Teatime's speech and mannerisms are never quite natural.
Poor Bilious. Sure, we don't feel quite as sorry for him as we do for, say, Rincewind, but the poor guy gets all the hangovers from the god of wine without ever getting to actually enjoy earning them.
There is a chance that since adult concepts such as death do not exist in the Tooth Fairy's world, then hangovers might not exist either.
Except Bilious gets queasy again just as he and Susan enter the building. If anything, alcohol may cause instant nausea there, for the same reason that the mooks couldn't smoke inside without feeling ill: because that's how a little kid who's sneaked into the drinks cabinet and tried some would feel.
The Cheerful Fairy. The Senior Wrangler, at least, falls for her because of her Woobie-ness. Then he becomes a Woobie when she vanishes just before he was about to get serious with her.
Cant Un Hear It: Regardless of your opinion on the TV adaptation, Michelle Dockery and Marc Warren's performances as, respectively, Susan Sto Helit and Jonathan Teatime are probably going to be definitive.
Foe Yay Shipping: Between Susan and Teatime. Apparently based on the confrontation scene in the Tooth Fairy's home which, with the right sort of squinting, can come across as sexual tension.