Central Premise Recycled From: "The Empty Child" mostly (see next point).
Reference to Moffat's Back Catalogue: WWII-set story involving small boys and their mummies, and something which looks villainous actually just trying to help out; parents as the real heroes; girls with pigtails; Christmas special which is a Doctor Who-styled reinterpretation of a British children's classic; the Doctor as some kind of wizard-figure who fixes everything for everyone. Though he's borrowed Gatiss' wooden dolls, and Davies' celebration of the nuclear family unit as some sort of ideal.
Amy Screws Up the day with Wuv: What's wrong with carol-singers, I'd like to know?
Robert Holmes Called...: It does make the story less saccharine knowing that the planet that's harvsting the trees is Androzani Major.
And from the Hiatus: There's a story in one of the Short Trips anthologies by Mark Michalowski entitled 'The Lying Old Witch in the Wardrobe'.
Murray Gold's Festive Number 1: None! What, did they run out of budget there as well? We couldn't have had a novelty Forties-style song number from Alexander Armstrong or something?
Nostalgia UK: The story takes place in that kind of fantasy WWII which lurks in the heads of the British, where courageous RAF pilots fight dastardly Nazis on behalf of stiff-upper-lipped mothers and their children, with none of the messy details like the Dresden bombing or black marketeering or Churchill's secret realpolitik or information censorship getting in the way.
Inside Jokes: Alexander Armstrong as a WWII pilot. Come on, who didn't think that his first words to his navigator would be "Vera Lynn, she's well fit, innit?" Rather a lot of Chronicles of Narnia inside jokes (Uncle Digby, sentient forests, a child's journey to an alternate universe providing a means of saving a parent, etc.). Androzani Major.
Teeth! None, they're trees.
Hats! A space helmet with airholes in the back, it seems.
Fish! I'd have to watch it again but there's got to be an aquarium in that playroom somewhere.
Small Child! Two of them.
Item Most Likely to Wind Up as a Toy: Bill Bailey's team would seem obvious but there might be some lawsuits from the designers of Halo over the look of their environment suits, so I'll suggest the big tree people.