Kate is wearing a very low cut blouse but the scar where she was shot isn't there.
When Castle and Beckett arrive in the Hamptons, the sun is just starting to set. It takes about two hours to go from Manhattan to the Hamptons, so it would have been at least 7:00 pm before they got there. The sun sets in October there a little after 6:00 pm.
When Beckett was watching the clock to turn 5:00, you can see the dial went by every second, meaning the second hand went from 1 to 2 to 3 and so on. After Castle arrived, you can see the clock on the computer screen is different. The second hand went continuously.
In the aerial shot of their car heading to the Hamptons, Castle is driving on the left side of the road.
When Castle and Beckett are sitting in the holding cell, Deputy Jones tells them that Chief Brady wants to talk to them. Beckett jumps up from the bench but is held back by handcuffs. She is not really cuffed to the bench but merely holding them in her hand.
When the deputy goes to unlock the cell to take Beckett and Castle to Chief Brady, Beckett jumps up forgetting she's handcuffed to the bench. When this happens, Stana Katic accidentally hits the cell door with her hand and it moves, revealing it was never actually locked.