Hogfather/WMG

WMGs for Hogfather. Warning: Potential unmarked spoilers.

Teatime's eye is missing because he cut it out himself.
Either he was planning from the beginning to replace it with a magic prosthesis or he was trying to emulate Odin Blind Io, as in this fanfic.

Young Teatime received a Red Ryder carbine-action 200-shot range model crossbow for Hogswatch and promptly shot his eye out.
It is A Hogswatch Story, after all.
 * He blames the Hogfather for it. That's why he came up with the plan to kill him.

The things popping up around UU was because of the High Energy Magic Building.
The Belief surplus only had an effect because of the HEM experiments making reality weaker (from the inside, rather than making it thinner like when the things from the Dungeon Dimensions can break through). Otherwise, it would have just made everything else (with the possible exception of Bilious) a little more real, rather than a few things around UU (and one thing where the Hogfather went missing) a lot more real.
 * In Soul Music, Ridcully mentions that wizards tend to be the first ones affected by weirdness like Music With Rocks In or the excess life in Reaper Man, so it might be more a property of wizards than the HEM experiments.
 * The series portrays Unseen University as stuffed with magic long before Ponder Stibbons graduated, so it probably isn't a consequence of the HEM experiments specifically. It does likely have a bit to do with reality being thinner at the university, though, just on a more generic level than what Ponder Stibbons has been up to. It probably is a combination of the high concentration of magic at UU weakening reality (easier for things to be brought into being) and the Soul Music-noticed tendency for wizards to act as canaries when it comes to reality getting wierd (odd - even non-magical odd - kind of things tend to happen to them early).

Nothing happened to Teatime's eye(s), he was just born that way.
Just saying.
 * He couldn't have been born with a glass eye. It's even mentioned (briefly) that he lost it in "some unknown accident", whatever that means.
 * That would raise even more interesting questions about that "special marble" that Gawain found in the fireplace.

Teatime's final intent wasn't to inhume Death...
It was to use the destruction of Death to get to Azrael, and truly kill death/Death.

The Little Match Girl died shortly after.
Death was cheating with presents by stealing things from other people; but "Hogfather magic" is determined by the monetary bracket that they are in. The Little Match Girl had no money. He CANNOT break the rules here. This is part of the series mythos itself, the only exceptions were the princess in Mort, who Death called in favors from all the gods (Something he couldn't duplicate for Ysabell and Mort in Soul Music), the girl in Reaper Man who Death exchanged his Lifetimer for, and Rincewind. None of these cases apply. There is no justice.
 * She died a few days later because of pneumonia. Perhaps Nobby was able to find some doctor (or vet) who was able to give some kindness to her before she died.
 * Granny Weatherwax might have been able to use a story to save her. (She did save that one baby from Death by roleplaying Elisha). Perhaps she could have made the self-righteous king give true charity instead of finding some peasant and shoving food into him. (and lecture him about the difference.) Death simply used the King as a Take That moment; which didn't accomplish anything.
 * Whether or not angels were there to comfort the Match Girl, her grandmother was. There is no justice. There is just us.