Does Ibn Hazm stipulate that the head covering be thin or thick, or waterproof, or worn inside or outside the home in order to wipe over it?

A thin garment that reveals the skin is not called a hijab, nor is it what Ibn Hazm meant when he permitted wiping over it. It is permissible to wipe over women's khumoor (head coverings) and their usual headscarves, whether a woman wears them at home or outside. Wiping is not invalidated by removing the item wiped over, according to Ibn Hazm. There is no harm in wearing the khimar for the purpose of wiping over it and then removing it.