The nonsense is mostly mathematics
Charles Dodgson lectured in maths at Oxford for twenty-six years and published on geometry and logic under his own name. Lewis Carroll was the pen name he used for everything else, and he disliked the two being connected. The jokes give him away regardless.
The Mad HATTER's tea party is a logic lesson in disguise: ALICE says she means what she says, and is told that is not the same as saying what she means. That is the difference between a statement and its converse, which is the first thing an undergraduate gets wrong. The CHESHIRE cat vanishing and leaving its grin behind is a joke about abstraction — the property of a thing hanging around after the thing has gone. When ALICE takes a POTION and starts to SHRINK, she worries about going out altogether like a candle, which is a child's version of a limit.
The riddle has no answer
Why is a raven like a writing desk? Carroll wrote it as a deliberately unanswerable RIDDLE and was surprised how much this annoyed people. Letters kept arriving. In a preface to a later edition he finally supplied one — something about the raven producing a few flat notes and never being put with the wrong end in front — while admitting it was invented afterwards and the question had originally meant nothing at all.
Other people have supplied better ones since — the neatest is that Edgar Allan Poe wrote on both — but none of them is the answer, because there is not one. It is a good thing to point out to a class: a question can be shaped exactly like a question and still have nothing behind it.
What Wonderland gave the language
Grinning like a CHESHIRE cat predates the book — Carroll took an existing regional saying — but he is the reason anybody still uses it. Down the rabbit hole and mad as a HATTER both attach themselves to him too, and chortle and galumphing are his outright, coined for Jabberwocky in the sequel.
Mad as a hatter has a grim explanation that is at least partly true. Hat makers used mercury to treat felt, and long exposure caused tremors, slurred speech and paranoia. The phrase was in use before Carroll wrote, so he did not invent the link, but his HATTER is the reason the trade is remembered for it rather than for hats.
Everything in the grid is from the two books
Wonderland supplies the white RABBIT, the DORMOUSE asleep in the TEAPOT, the March HARE, the DUCHESS, the GRYPHON, the QUEEN of HEARTS playing CROQUET with a FLAMINGO for a mallet, and the KNAVE on trial for stealing the TARTS. The CURIOUS ALICE, the shrinking POTION and the MAD tea party are all from the first book.
The looking GLASS belongs to the second, Through the Looking-Glass, published six years later and built on a chess problem rather than a pack of cards. Tweedledum, Humpty Dumpty and the Jabberwock are all from that one, and plenty of people who are sure they have read Alice in Wonderland are actually remembering it. The stories have been filmed more than twenty times, including by Disney in 1951, which blended the two books together and made the confusion permanent.