For some scenes, a "jelly" or a fake baby was used as a stand-in for Presley Smith (Sunny Baudelaire). This is mostly due to the fact that they couldn't put an actual baby in a cage. Presley was photographed, and when the prop baby was ready they sat the baby actress next to her stand-in. She did a double take and laughed very hard to see herself like this.
The hook-handed man tells Count Olaf that the cake is a little "lemony" and count Olaf says back "I told you never to say that word." This is a reference to Lemony Snicket, the pen name of Daniel Handler, the book's author.
According to book and series writer Daniel Handler (aka Lemony Snicket), Neil Patrick Harris (Count Olaf) added the "I bought it online" line to the hourglass scene.
Lemony Snicket's photo of the trolley has a note attached to it that says, "The cure for anything is salt water - tears, sweat, or the sea." It's a quote from gothic author Karen Blixen.
When Sunny goes missing, Count Olaf asks "When did you see her last?" This phrase was the name of the second book in "All the Wrong Questions," a prequel series to A Series of Unfortunate Events.