The Valonqar

I dont know. There was this over lay of clashing steel swords there, as the proud smile slowly fell from his face that made me think that while, yes, he was realizing he has set his daughter on the path of a warrior. - But that is a realization I imagine he had with Rob and Jon, and would have had with Bran and Rickon as well. He was more having flash backs. Personally, I think they were flash backs to the Tower of Joy. After all, Arya is supposed to look like Lyanna, and Ser Auther Dayne could weild a sword with one hand, hell, he could duel weild. Which I bet is extreamly unheard of in Westeros to wield two full blades at one time 

It could also be the realization, as he watched Arya rapidly improve in that scene, that Arya's path would be a difficult one, because clearly, the gods had fashioned her to be a swordswoman, and swordswoman did not have a place in Westeros society. 

As for what her end game could be if she doesnt kill Cersei... arg. idk. I like the leaks hinted that she goes back east. Maybe she returns to the Faceless Men in the end. Her list has been completed, she has had her vengeance, given the names to the god of death and now, with the remains of her family safe, she is content to be No One, since she has no place in Westeros.