Controlled experiments are very rare if you are talking about gravity, since the force is so weak. Observational experiments (astronomy) is where most of the evidence comes from.

The most obvious inclusion of it for me is the radiative (also known as the photon) component of the matter density in Lambda CDM models (that is, they need to account for the proportion of the energy density of the universe which is related to electromagnetic fields to rule it out as dark matter).

I'm not sure what you're looking for. EM fields so strong that they affect spacetime noticeably?

We haven't observed any Reissner-Nordstrom/Kerr-Newman black holes iirc. That said, _anything_ with stress-energy-momentum is held to affect spacetime curvature in GR.

I think there are no experiments in the lab or a direct measurement of this. Anyway, if the light is blended during an eclipse, the that light should have accelerated the Sun just a tiny tiny ... tiny tiny bit in the opposite direction.