" . . . what happens if the force is also acting in higher order dimensions? What if the originating force is being projected on a higher dimensional surface area? Are there forces which vary to other powers than the inverse square law?
Yes, there are. The Casimir force related by the above equation is known to vary as the inverse d4, keeping all other parameters constant, which is two orders of dimensions higher than the more common inverse square law forces, and coincides with a force projected over the surface area of a 4D hypersphere (see table above). Such force that varies with the fourth power of the distance can be thus considered as a force vector that is scattered in a 4-dimensional space. Therefore, it is evident that the field that originates the Casimir force is a 4-dimensional field, that it is in fact a hyperspace field that produces the corresponding effects in our restricted 3D vision of our universe.
So, monitor a specified (inert, very stable, large atomic weight) molecule in a protected environment for variations in F.