Space-time curvature caused by electromagnetic fields - an experimental results
The paper requires a Fabry-Perot cavity that can store light for 200 days, and then measure the phase to a trillionth of a wavelength. This is far beyond the state of the art. With 100 meter-long arms, it would require mirrors that lose less than a quadrillionth of the light with each bounce. To maintain relative coherence between the two arms, you would need to have exactly zero atoms of residual atmosphere in the beam paths, and the two arms would have to be equal in length and stable to about 10^-33 m (about a hundred Planck lengths).
This experiment has not been done.
However, you can show that electromagnetic fields have gravitational effects. A small but significant (calculable, measurable) fraction of the mass-energy of an atom is due to the EM fields in the nucleus and between the nucleus and electrons. If this field mass-energy did not have gravitational effects, then the gravitational mass (measured by weight) of an atom would be different from its inertial mass, which is not the case to as accurately as anyone has been able to measure. (Any discrepancy would be an instant Nobel. Google 'Equivalence Principle' for more information.)
For your third question, gravitational waves are an example of gravitational fields having gravitational effects, in the same way as EM waves are an effect of the electric field having magnetic effects and vice versa.
Also, the precession of Mercury can be thought of as gravitational fields having gravitational effects. When Mercury is closest to the Sun, there is less gravitational field within the sphere with radius of the Sun-Mercury separation. Since gravitational field energy is negative (surprising but true), this means that Mercury feels more gravitational pull close in than would be predicted by a Newtonian 1/r^2 field, leading to orbital precession.
Another example is given by a gravitational wave detection that was also seen in electromagnetic radiation (a hard X-ray/gamma burst). The gravitational field along the path, caused by clusters of galaxies, etc., results in significant propagation delays for electromagnetic radiation compared to the Euclidean light speed calculation. (We know that this effect is real and consistent with theory for EM waves, because that's how gravitational lenses like the Einstein cross work.) The EM and GW detections were ~simultaneous, indicating that GWs are delayed in exactly the same way as EM waves.