Lords of Winter Edd Stark has LoW Shaggydog and Crown of Winter attached. The other player wants to attach 'bastard' to Edd. Edd is immune to character abilities, but 'bastard' isn't a character and besides Edd is immune, Crown of Winter isn't .

So Edd should lose both his attatchments, right?

Yes. As you say, Bastard is an attachment. Something would need to be "immune to attachments" to be protected from it. Further, a character's immunity does nothing to protect the attachments on it. Immunity only protects from things that try to directly affect the immune card - and losing an attachment is an indirect effect at best.