Nope. Tywin knows she's a northern girl, if he knew she was high born as well he'd find out who she was and pack her off to King's Landing as a hostage given that all the North lords are in rebellion. My guess is he suspects but has doubts - he'd look daft in front of his men if he thought a commoner was high born so he's keeping her close and engaging in verbal duels with her. Arya mis-speaking, saying "proper" when she should have said "properly" could well have saved her.
hmm, judging by the fact that he oversee's a bunch of very violent torturing men. If he thought for a second, surely he'd just stick her on a rack?



