My biggest problem with Tywin & Arya is that he obviously knows she is educated and likely highborn. He knows she is lying to him. The only conclusion he could draw is that she is associated with his enemy. That puts him in danger, she could be a spy or as was suggested in the show, might try to kill him. Additionally, holding the daughter of potentially one of your enemy lords could open some war strategies (ransom, blackmail etc.). Despite all that, the REAL Tywin probably would have just executed her for lying, or tortured her to find some intel. And WTF was up with Littlefinger being there? Are we supposed to thinking he recognized her there? No idea how that will impact either of their arcs..
Also, Jaqen H'ghar doesn't really operate like a knight in shining armor, as his emergency assassination of Amory Lorch (sp?) (whom she never had killed either) would indicate
. Furthermore its not really arya-like to do something as stupid as steal letters in plain sight, carry in plain sight, and go running for help.