Tywin was never in Kings Landing during the time of Ned being in Kings Landing with Arya and Sansa.
Tywin was fighting the war when Ned loses his head.
Tywin was still fighting the war when Arya enters Harrenhal.
Tywin has shown no evidence of visiting the Starks prior to Game of Thrones starting.
----------------
Arya knows who Tywin is because he leads an army and she is told he is the head Lannister, Tywin does not know who Arya is for he is told she is a mere serving girl.
----------------
Yes the house of the Undying sucked, but then again they also never had the tower of joy scene.
He would have gone to see the leader of another important kingdom and he would have visited and seen the child. He also would have seen a portrait and I thought at one point that he had visited the castle during the books but I may be wrong there (an excuse to reread I guess 
). I also thought Arya recognized him on site before she was told who she was for similar reasons as I expected him to know her.
Still you raise interesting points that I had not considered.
