Can gravitational waves diffract off massive objects? Can gravitational waves diffract off massive objects?
If so, how much mass is needed to significantly disturb the waves propagation?
 A: It seems to be still a research question for specific situations , but the answer is yes, gravitational waves diffract. example this paper:
Emission of gravitational waves from binary systems in the galactic center and diffraction by star clusters
From the abstract:

The diffraction pattern of gravitational waves emitted by a binary system by a cluster of stars has been also analyzed. We remark that this is only a preliminary-theoretical work than can acquire more interest in view of the next-coming gravitational wave astronomy era. 

in the paper:

it suffers diffractive effects while interacting with stars. Hence, by considering each star as a circular slit and applying the well known theory of wave diffraction, we can evaluate the expected diffraction patterns
  on the observer plane.

Other  papers studying diffraction of gravitational waves here and here.
A: When light enters a media it gives some energy to its charges and there is a reemission that, added to the initial wave, makes the velocity change, makes the refractive index value.  To stretch and expand the earth when passing, in order to move the mirrors of the Ligo interferometer, a similar phenomena may exist. And, of course when traversing other objects in cosmos.
A: Diffraction is bending (turning) around the corners. The turning is not possible without having slow speed at the edge, and faster speed around it.
For gravitational waves, there is no known theory that says that GW speed changes through (or in proximity of) any matter. 
Therefore, no diffraction of gravitational waves.
Even if diffraction is due to reflections caused by the irregularity of the edge, still no diffraction for GW as they are not expected to reflect off anything except probably, the singularity.
Thinking more on the question - Diffraction is a phenomenon of light where it bends around corners (i.e. obstacles). There is no known normal obstacle for gravitational waves, because, they pass through everything. Therefore, the concept of diffraction itself should not apply on gravitational waves. While saying "no obstacle", I have excluded complex entities like black hole/event horizon/singularity.
