Classical civilizations used a rounded arch, not a caternary arch, which only started to been used in Europe in early modern times.

When the igloo was invented however, we don't know because, as you mentioned, there is no archaeological remains of igloos themselves. The Dorset culture lived in the arctic before the Inuit and there are a couple of artifacts from them that have been interpreted as snow knives, which are needed to build igloos. The Dorset culture starts at ~500 B.C. It has been suggested that igloos are far older than that and that their history stretches into the Pleistocene, but there is not hard evidence to support this.