Is self gravitation theoretically impossible? Is it theoretically possible to create some system such that the energy distribution creates a gravitational potential offset from its center of mass (or energy?) so that the body continually 'falls' into its own potential (i.e., as the body moves, the center of the potential moves with it such that it constantly falls forward)?  I get the feeling that this doesn't make sense in the context of GR so I'd like to know why (I'm trying to understand the frame dependance of gravitational effects).
 A: In Newtonian mechanics the motion of the center of mass can 
not be affected by internal forces. In particular, if the center of mass is at
rest, it can only be moved by external forces. 
However in curved spacetime this situation changes and an extended body can shift its position through the use of internal forces only. This fact was first noticed by J. Wisdom:

Wisdom, Jack. "Swimming in spacetime: Motion by cyclic changes in body shape." Science 299.5614 (2003): 1865-1869. (doi:10.1126/science.1081406) (online version)

From the abstract:

Cyclic changes in the shape of a quasi-rigid body on a curved manifold can lead to net translation and/or rotation of the body. The amount of translation depends on the intrinsic curvature of the manifold. Presuming spacetime is a curved manifold as portrayed by general relativity, translation in space can be accomplished simply by cyclic changes in the shape of a body, without any external forces. 

This result produced some followup studies (Google scholar).
Notice, that the translation appears as the result of a series of deformations of the body (through the use of some internal forces, nongravitational in origin).
Another notable point, is that such translations at the reasonable macroscopic conditions are very small: Wisdom has estimated that a 'swimmer' near the surface of the Earth using meter-sized deformations can achieve a displacement of $\sim 10^{-23}\text{m}$. M. Longo improved  this to $\sim 10^{-7}\text{m}$. Though satellites with masses gyrating on many kilometer long tethers may generate translations which are quite noticeable.
