Do gravitational waves slow down as they pass through matter? I've heard that gravitational waves travel at the speed of light, and have some parallels to electromagnetic waves.
EM waves slow down as they pass through matter (speed of light in glass is slower than in vacuum, for instance).  Do gravitational waves also slow down as they pass through matter?
If so, are there any effects like Cherenkov radiation when matter passes through a medium at a speed greater than the velocity of gravity in that medium?
Do large masses like stars or Jupiter act as ball lenses for gravitational waves?
 A: Unlike electrostatic charges, mass is always positive, so that there is no dipole density to deflect gravitational waves as they pass through a material. So the answer is no, not in an analogous way.
The main effect is the gross retardation of gravitational waves by matter the same way that they retard light, by focusing. This is just the bending of light/gravitational-waves associated with gravitational lensing.
A: You ask:

Do gravitational waves also slow down as they pass through matter? 

and an answer has been provided above.

If so, are there any effects like Cherenkov radiation when matter is traveling faster than the speed of gravity?

To be clear the speed of gravitational waves is supposed to be  c, the speed of electromagnetic light.
The recent publication of the OPERA superluminal neutrino results  brought out a study by Andrew G. Cohen, Sheldon L. Glashow that answers the question positively. Matter moving superluminally is expected to  radiate a ring of pair production about its path, according to them.
