You seem to be describing [Lev Grossman](https://en.wikipedia.org/wiki/Lev_Grossman)'s [*The Magicians*](https://en.wikipedia.org/wiki/The_Magicians_(Grossman_novel)), and at least one of the sequels ([*The Magician King*](https://en.wikipedia.org/wiki/The_Magician_King), and [*The Magician's Land*](https://en.wikipedia.org/wiki/The_Magician%27s_Land))

[![Cover of Lev Grossman's The Magicians][1]][1] [![Cover of Lev Grossman's The Magician King][2]][2] [![Cover of Lev Grossman's The Magician's Land][3]][3]

  * The main character, Quentin Coldwater, is a brilliant emo kid who is really good at (stage magic) card tricks, and who is obsessed with the magical land of Fillory (a kind of grimdark-ish/*Less Than Zero*-ish version of *Narnia*). He is recruited into Brakebills College for Magical Pedagogy along with many other kids. His best friend is also recruited, but fails the entrance exam.

  * This was a text novel, not a picture book. That said: there is a [television adaptation](https://en.wikipedia.org/wiki/The_Magicians_(U.S._TV_series)) that has produced four seasons of material loosely based on the books, so there are some visual representations of the books.

  * Quentin, his best friend Eliot, and their other best friend Janet to indeed hook up. That's one of many configurations of Brakebills students and other characters hooking up in the series.

  * I do not recall the specific line with the talking Rabbit, but Fillory, which Quentin and friends (in various combinations) do spend time adventuring there across the three books, is simply infested with talking Animals. They mouth off, and generally have earthier personalities than the twee characterizations of talking Animals in C. S. Lewis' *Narnia* books.

  * One of the baddies in the books, does indeed show up, and bite off the hands of one of Quentin's classmates (Penny's), which present difficulties in Penny being a magician, and driving him towards a certain shit in career aspirations.

  * Quentin and friends do indeed go hunting one of several magical beasts (can't recall if it was a white stag, but that sounds about right), which can grant wishes in one of their Fillorian quests.


  [1]: https://i.sstatic.net/ulMoim.png
  [2]: https://i.sstatic.net/Sk8oZm.png
  [3]: https://i.sstatic.net/SjPEQm.png