Towards the beginning, it is shown that Castle has been away for 2 months. Thus, as the wedding was in May we can expect that his admission to hospital would be sometime in July (or early August at a push). However, his wristband in the hospital reads that his date of admission was October 2nd 2014 (10/02/2014). Whilst this brings Castle up to 'real time' it means that he was either gone for over 4 months or that there was a prop error.
During the flashback of the accident, a 2005 model Mercedes S class driven by Castle is shown beside the black SUV. Immediately after, at the accident site, the destroyed vehicle is shown to be a 1999 Mercedes S class which is of a similar color but looks nothing like the car Castle was driving before the crash.
When Beckett takes Castle to the place where he allegedly had lived in a tent while he was missing, Castle tells her he could never have stayed there because it faced the ocean and, due to research he had done for an aborted book, he had an abiding fear of tsunamis and said his house in the Hamptons was on the inland side, not the ocean side. However, in "A Deadly Game" (season 2), Castle tried to convince Beckett to come to his house in the Hamptons because it was right on the ocean.
In previous episodes of Castle, we've seen his house in the Hamptons along the beach. He tried to convince Beckett to come with him, promising a beautiful beach view. However, in this episode, Castle says that he bought the Hamptons house inland to avoid being next to the water. This makes no sense.
The U.S. Coast Guard finds him adrift in a small boat off the coast of Delaware. But CGC Narwhal WPB 87335 (you can see the hull number as they pull up to the dingy) is stationed out of Newport Beach, CA. And a WPB would never be on patrol that far from home port.
The fire department sprays water from the bridge to put out the blazing car. When Kate realizes Castle wasn't there, she says, 'Even with a fire that hot, there would be a body.' While examining the car, Esposito touches the car, which would have still been extremely hot that soon.
When on Long Island, it can be seen that there are hills that rise several hundred feet. No such heights exist on Long Island.