At 26:33 when Esposito first comes in to reveal that he talked to the credit card company, his badge has slipped down from the holder. Seconds later when he brings out the note paper with the address he tracked down, the badge is properly seated in its holder.
Becket says one of their suspects had previously been convicted of manslaughter for trying to run over her boyfriend with a car and had been sentenced to five years in prison. Manslaughter requires you actually murder someone, not attempt it. The charge would have been attempted murder.
Castle's decision to spend the summer in the Hamptons to finish his manuscript happens just before Memorial Day. This episode takes place on the autumnal equinox which is usually on the 22nd of September. It takes many months for a manuscript to be published. Even if he had finished and turned it in before Memorial Day, it still wouldn't be ready for release before Christmas.