Stupidity: When Castle and Beckett discuss the trap LokSat set for them at the dead drop and any forthcoming retribution, Beckett comments that her dad should be relatively safe from reprisal because he's on business out of town. Someone as smart as Beckett would never make such brain-deadly dumb assumption; considering how powerful and how dangerous he's supposed to be, especially with the resources of the CIA at his disposal, LokSat surely would have no trouble finding and getting to, and certainly no qualms about eliminating, Beckett's dad if it stopped her in her pursuit of him.
Stupidity: When Sonia cuts off her monitoring anklet in her attempt to escape, Ryan and Esposito try to hide the fact from Beckett, an idea that, for the two of them, is absurdly stupid. With at least a decade of experience each, they're too smart to not know that (A) most, if not all, monitoring anklets are tamper-proof (if they weren't, what deterrent would stop criminals from doing exactly what Sonia does and escaping?), and (B) as captain of the precinct who requested her furlough, responsibility for Sonia ultimately falls on Beckett and she would have been notified by the security company the moment they knew that the anklet had been tampered with.
Stupidity: Castle casually smashes two dolls expecting there to be something hidden inside, but a) he knows they're valuable collectibles which the captain would take serious issue with him smashing, and b) anything inside must have gotten inside somehow non-destructively, so there's no need to smash the dolls, why not get whatever's inside out the same way it got in?
Stupidity: The dancing show is appallingly organised - they launch into the opening dance without any of the production crew knowing where one of the two dancers is, leaving everyone mystified when she misses her cue (because she's dead in her dressing room). They wouldn't even have started filming without her being offstage ready to go.
Stupidity: Castle and Beckett are trapped in a refrigerated container - they bang against a steel door and shoot it, but never even consider doing anything with the refrigeration coils or vents which are visible, accessible, and what keeps the container cold. Damage them and the space they're trapped in wouldn't be below freezing any more, which is the main danger to them.
Nanny McDead - S1-E2
Stupidity: Mrs. Peterson tells the cops in the first interview that the phone went 'straight to voicemail', which would happen if the phone was off, and it's not the case. There's no reason why she'd lie about this particular fact. Which ties into the larger problem that the cops are looking for the missing phone but nobody tried to call it nor the phone company told them it was still on, or the plot wouldn't work.
Answer: The networks are now having winter breaks. A mid season end of new episodes, when they come back after the new year, it picks up as if several months have gone by. Beckett suggested Valentine's Day as a "chick flick" to watch as she is moved in. Also some episodes are not scheduled to air in order.