Fin pulls out a library card from a book that Olivia had checked out in her name. In the old library card/check system, the card would have been kept at the library to keep track of who had checked it out. It shouldn't have been placed back in the book, then released to check out.
When Fin is selecting the train book in the library, he pulls it from a shelf that contains fiction, mostly James Patterson novels. Fiction and non-fiction books would never be on the same shelf, even in a very small library.
After Olivia returns home from the hospital, a brief shot shows Fin and Olivia watching the sunset from her dock. It's actually from an earlier scene in which Fin and Olivia had a conversation on the dock. They're wearing the same clothing.