I thought photons also contributed to the curvature of spacetime because they have momentum (and thus energy). I would expect it to be a pretty miniscule contribution, though.
However, whether that translates to a loss in momentum is a bit more fuzzy and I really can't tell whether it does or doesn't. Although I'm far from being a physicist, so hopefully someone more knowledgeable chimes in to enlighten us...
But I'm curious: what is your reasoning for asking whether photons have a mechanism to lose momentum as a consequence of them affecting the curvature of spacetime? It's not at all obvious to me the relationship between these two concepts.
This is not true, elastic scattering is very common.
Yes, redshifted photons lose momentum. Momentum (really the stress-energy tensor) is conserved locally, and along trajectories that preserve the metric, but global momentum conservation in GR isn't even well-defined.