Character mistake: Castle claims that they are technically standing on Russian soil inside the consulate. Contrary to popular belief foreign diplomatic missions, while being afforded special privileges (ie. Diplomatic immunity), are not considered sovereign territory of the represented state. Because of that Castle and Beckett are in fact on U.S. soil.
Continuity mistake: Beckett keeps a jar of nuts on her desk. When she's talking to Castle about Lars, she reaches over her desk to put the jar down. But in the following shot when Castle stands up, both her hands are suddenly in front of her laptop, clasped together.