Just slightly confused about the house icons, such as the direwolf head for Stark. An attachment that says "direwolf head" characters only, such as Ice, can only be attached to House Stark characters.

I'm assuming "nuetral characters" aren't House Stark characters, even though they are in a Stark specific deck? For example, the Jon Snow card from the core set has a Nuetral shield, does that mean he can't have ice attached to him?

Correct. Neutral cards have no House affiliation, period. So things like Ice, which are limited to characters with a particular affiliation, cannot be attached to them.

Neutral cards do not "take" a House affiliation from the House card of the deck they are in.