As said, non-stationary, or non-asymptotically flat spacetimeI can't imagine a situation where no definition is possible.
Yes, but in GR there is no global refence frame!That sounds like another definition issue, and I think even a wrong one. When comparing energies one must use the same reference system
You have local frames which are not related a priori and which do not allow for a global definition of energy. In order to compare energies you have to define how to transport reference frames through spacetime (along null lines). But once yo have done that you get the redshift for free - w/o defining energy.