In the beginning of the episode, Beckett claims that Russian consulate is legally considered foreign territory. In fact, under the Vienna convention, the consulate (embassy) is on American soil, it is just excluded from US jurisdiction (cannot be searched, seized etc.)
The assassin has false IDs made so he can get into the consulate, but he shoots at his target from a building across the street so the false ID was not necessary.
Rita tells Castle that his father never talks about him, but short time later she says his father is very proud of him. If he has never spoken about Castle then there is no way Rita would know that. In fact, since we know that his father has hidden the fact that Castle is his son, she wouldn't even know he existed.