Did Tywin know it was Arya at Harrenhal?

He never would have passed up the chance to use her as a hostage or bargaining chip against Robb had he realized who she really was. He may have deduced this after she left Harrenhal, but he never would have admitted this since it would have made him look bad.

Did Tywin know Arya was a stark?

Due to this, Tywin deduced her to be a high-born, even a Northern one (probably of some lesser lords). He found her attitude amusing, even comparing it to his kids, but he would have never imagined it in his wildest dreams (like any normal human being) her to be the Arya Stark.

Did Tywin know Arya was not who she claims to be?

There are many ways Tywin could have spun having possession of Arya Stark and emerge from all of it smelling like a rose. You know @Maxxine, Tywin knows Arya isn’t who she claims to be and I found it very odd that he just let her be.

Was Tywin ever suspicious of Daenerys’ true identity?

Some fans have wondered if Tywin was ever suspicious of her true identity and realized she was a Stark, but this is unlikely. He never would have passed up the chance to use her as a hostage or bargaining chip against Robb had he realized who she really was.

Could Tywin have demanded to see Jamie at the trade off?

Tywin could have demanded to see Jamie at the trade off. It’s not like Robb could actually just kill Jamie right then and there–Sansa’s still hostage. He could claim he had Arya all along or that she was sent to the Westerlands immediately following Ned’s execution.

What should Tywin do about Harrenhal?

Tywin has a war to fight, and Harrenhal is a military garrison. At that point he was still on good terms with Cersei and Tyrion, so let them handle it. Remember, he has appointed Tyrion as his acting Hand, so Tyrion is the one in charge of non-military matters at this point.