On what basis did the faceless request the waif to serve them ?

[Gneisenau]

The FM take something dear from the person requesting a kill.

So the Waif's dad paid his wealth and his daughter, but this would imply that he owned his daughter, aka some form of slavery.

What basis do the FM for requesting the waif to serve them ?

[EastTymor]

To put it in modern terms, the waif had not yet attained adulthood and her father was her legal guardian, Given this, it was his right to to decide how she would be brought up, educated, etc (think sending a kid to boarding school). Alternatively, he transferred legal guardianship.

More alternatively, if you are a cult of unstoppable assassins you don't need a basis to do what you want.

[lomiller]

The FM take something dear from the person requesting a kill.

So the Waif's dad paid his wealth and his daughter, but this would imply that he owned his daughter, aka some form of slavery.

What basis do the FM for requesting the waif to serve them ?

The price was his sacrifice not her service so she was never a slave but she was their to raise and train as they saw fit. Presumably she would have been given the same types of choices Arya was given when she was old enough to make them. Remember going back to her family was not one of the choices they offered.