Are gravitational waves affected by gravity? When a gravitational wave passes by objects, do their gravitational fields warp or change it?
Gravitational waves are just the way a change in a gravitational field is communicated. If the change oscillates, you get oscillating wave forms like we see from binary systems that make up the waveforms we all know and love.
Since these waves are (according the the GR way looking at gravity) oscillating propagating parts of spacetime, they propogate through other parts of spacetime such as other curved spacetime. The geometry of the curvature of the change in medium does (contrary to what I previously thought) affect the gravitational waves; one example of this the redshirting of the gravitational waves when they pass through a gravity well. For similar reasons of light, the medium itself is stretched and curved causing gravitational redshift in the gravitational waves themselves. Furthermore, if space is curved, naturally the gravitational waves’ direction will be affected, but in a more complicated way than just curving like a photon since they can interfere with each other and the differences in curvature from one spot would lead to some complicated behavior and nonlinear interference patterns.
In short, yes, gravity does affect gravitational waves, and probably in more ways than listed here.