Yes, being waves they should interact with each other, just as any wave does. So they should be susceptible to interference and diffraction for example.
Everything does indeed produce these waves, but the intensity of these waves are very small (which is why we have had trouble detecting anything to date. So at low (quiet) intensities there will be quite a bit of noise. However, this result is "clean" because it comes from the "loudest" source of gravitational waves, namely the merger of two extremely dense entities. Think of it as a rogue wave on an otherwise placid sea surface, or a supernova in another galaxy shining out against its background stars.