Remember they met when HE was in Westeros, so I don't think they have any territorial restrictions. Not so sure about her reclaiming her name/title, but it could be rationalized as one of many faces that she now has.
In terms of her being allowed to live, my interpretation is that when a name is given to a faceless one, they are obligated to attempt to deliver it, but the Faceless God isn't actually particular about which face is received. Arya dispatching the Waif and bringing in her face settled things from each perspective. Not sure if that makes sense, but not much from their Order does.