1. The entire "Bones of the Slain Dragon" set (including the sword Edge of the Slain Dragon not shown in the screenshot)
and Ember Edge (new Frostivus sword) do not have the invisibility effect.

2. Why are all other skirts visible from the front in loadout except Cape of the Drake?
It can't be optimization issue because it would surely apply to other skirts as well. Wouldn't it?
In the screenshots you can see all the skirts compared (including default one):

3. Light Cape of Sir Davion is visible from the front in loadout, but it's not visible in-game.