When Beckett and Castle walk through the hotel room door the door jamb can be seen to have no damage despite being kicked in a few seconds earlier.
Euros found in a car at 8:03 (banknotes of 50 Euros) are obviously fake. Real 50 Euros banknote is of far brighter orange color.
Demming tells Kate that the little place around the corner from their beach house just had a reservation open up. If they have a beach house, there would be no need for a reservation around the corner. Also, they obviously already had plans to go to the beach house, so his last-minute announcement that they could go does not fit. Her conversation with Castle confirms the fact that they had prior plans to go to the beach house that weekend. However, the place around the corner could be a restaurant that they want to go to for dinner while they are at the beach house.
Beckett kicks in the hotel door causing damage even though there is a hotel employee present who could have used a pass key. Kicking the door in is very dramatic but no police officer would do that when there is a better way to gain entry that does not make the department liable for damages.
Everyone calls the party a "surprise" party but Castle knows about it so it isn't a "surprise" party.