Is it possible to build a multi-mode "fiber" with an air core so that a significant part of the light's path is through air? The outer sheath made of optically dense material would only be used for internal reflection to "guide" the beam...

No. The core has to be denser than the outside for the guiding to work; if you made a "fiber" like you describe the signal would just travel through the sheath - if it was heading towards the core it would internally reflect to stay within the sheath.