Formerly non-removable housing items.

[Acilith]

Does this also extend to say changing a carbuncle house to a different house skin?

Yes, I took off my paissa skin and it was in my bag. However, you'll need each piece of the exterior if you plan to go default and not use another skin. So you'll need a door, a window, walls and a roof. No need for the gate, unless you want a nice gate.