Why is Hiiro a Nora? We know that she has spent a lot of time with Yato (because they've shown to be together since childhood and Hiiro was his first Regalia) and also really cares about him. Yato himself says:


  


This means that Hiiro does care about Yato and has never done anything which she thought wasn't for the good of Yato. Just before this scene, we also see Hiiro saying that she wouldn't mind dying in the underworld as long as Yato was with her.

So why is it that she is a Nora? Why did she allow other Gods to name her?
She's first named by his father (the "sorcerer" is mentioned as her true master in the wiki so I suppose that means he named her first) as Mizuchi and then his father let Yato name her as Hiiro.

I think she never tainted Yato  because she cares for Yato so much that she never did anything a regalia shouldn't do to taint their master.

I haven't read that far in the manga so I can only suspect that the sorcerer allows her to be named by others for some sinister plan against the heavens.

http://noragami.wikia.com/wiki/Nora