It seems to me that Tywin knows she's lying about where she's from. He doesn't know she's Arya Stark, though. And he's enjoying the fact that he's got someone smart to talk to, but he also doesn't like it when someone gets off a good one at his expense.
Sansa passes the Hound in the halls of King's Landing. She thanks him for saving her from those would-be rapists and calls him brave. He says it's not brave for a dog to scare off rats. He says he enjoys killing, and that it's the sweetest thing there is. Even Ned Stark liked killing people, he says. Sansa doesn't like the sound of that. The Hound says she'll be thankful for him someday, when she's queen and he's all that stands between her and Joffrey.
Now, let's go check in on Daxos and Daenerys in Qarth. He's called a meeting of the Thirteen to look into the important dragon-stealing issue, but he assures her that her dragons mean nothing to him personally. She says they're more valuable than anything in the world. He points out that he's super-rich already. He keeps talking about all the riches stashed in his vault, but we've never seen any of it. She asks why he would help her find her dragons, and he says he's the one that invited her into the town. He needs to back up his words because if people call him a liar, he's nothing. She doesn't care about any of this. She just wants to stomp around and shout about her missing dragons.
Back to Snow and Ygritte. Trudge trudge trudge. Oh, how they trudge. She says the Free People may not live in castles, but they're free. And they don't serve a king who is only king because of his father. Snow objects that they all serve Mance Rayder, but she says that the difference is that they chose Mance. She encourages him to leave the Night's Watch and feel the glory of sleeping in and having casual sex with people like her. But even the enticement of girls clawing each other's eyes out to get at him won't do it. He says he knows how to have sex, and she tells him he knows nothing. Actually, that line seems to be a big touch point for fans of the books, so I guess I should recap it in full. So here you go: "You know nothing, Jon Snow," she says.
Hey! It's Robb Stark! He's holding court in his tent (which could be called "holding tent," I guess), finding out what Cersei thought of his offer of peace. He's talking to a Lannister by the name of Ser Alton, who doesn't want to say, but eventually admits that Cersei tore the paper in half. Robb magnanimously announces that Ser Alton's pen should be cleaned, so he can be comfortable in his hostagehood. Unfortunately, it's full. And so are all the other pens. In fact, they have so many prisoners that they no longer have room to lie down. A boy named Torrhen is told to take Alton to Jaime's pen. Well, it's called "The Kingslayer's Pen," which sounds a little classier. It's like the Presidential Suite!