I think the issue is that your noise texture has too much contrast in it. Both, displacement and bump loves smooth transitions between high and low values, but the former can still produce usable, albeit sub-optimal results with super contrasty textures, while the latter can't do that. It's the nature of bump mapping.
As for the projection issue, i think it's your mapping at fault, not Corona. It looks that currently your object has simple planar mapping. Give the cookie proper UV unwrap and it should render just fine.