Not if Aizen didn't know his name. Maybe he only knew what Nero told him his name was. Nero probably told Aizen part of his full name since he didn't trust the man as a Vasto Lorde.

As for the Gigai... the wiki page doesn't mention if a Gigai can die or not. I know they can get injured and bleed though since Rukias Gigai was seen injured at least once. Therefor it is my belief they can be 'killed'. The spiritual entity inside it wouldn't be harmed but I'm just making an assumption.