If gravitational waves experience gravity, that means that gravitons don't just interact with the energy-carrying particles of the Standard Model, but there is a graviton-graviton interaction as well. Two different gravitational waves, in Einstein's relativity, should interfere when they meet.

Are gravitons responsible for gravitational force?

In theories of quantum gravity, the graviton is the hypothetical quantum of gravity, an elementary particle that mediates the force of gravitational interaction.

Are particles affected by gravity?

The weakest, and yet the most pervasive, of the four basic forces is gravity. It acts on all forms of mass and energy and thus acts on all subatomic particles, including the gauge bosons that carry the forces.