Redshift changes the energy per unit of time, but not the total energy. A single photon from a star coming towards us will be 'blue shifted', but it can contain the same energy as a single photon from a star receding from us. It will take longer for the red-shifted photon to transfer all of its energy than the blue-shifted photon, but if they were equivalent photons when they were created, then their total energy will be the same at the end.
No, that's the wrong explanation. The time it takes for a photon to transfer its energy is independent of its (center) frequency, but depends on the envelope of a photon. You can reshape the blue-shifted photon so that it takes forever to transfer its energy without changing its energy. The bandwidth of a photon is related to the time it takes to transfer its energy and not the energy itself.
Energy is not invariant during a Lorentz transformation. This should become obvious very quickly: An object at rest has no kinetic energy. But if you look at it in a reference frame that is moving relative to it, the object suddenly has kinetic energy. So the answer to the question is, that you are comparing apples to oranges as the energy in the reference frame where the emitter is at rest is something different than the energy in the reference frame where the absorber is at rest. So to convert the energy from one frame to the other you would need to make a Lorentz transformation, which then would give you the energy of the red-shifted photon.