I’m 0% an expert in this, but I think they move at light speed all the time. Light is “affected” by mass only indirectly, since the light travels in a straight line through local space but space itself is curved by the mass.

Light gets caught up in mediums because those mediums have electric fields (the electrons for matter, or light itself when interfering). Thus, gravity waves will be slowed by gravity fields, like planets, stars, and galaxies.

What waves interact with also depends on the wavelength, like how radio waves can bounce off Earth's ionosphere, but can ignore the atoms in the walls of your house. There are plenty of very large gravity waves from merging black holes and neutron stars, and those pass right through Earth. Smaller gravity waves (like from a collapsing or disappearing star) could interact with other stars, possibly reflecting off of star clusters, or even refracting through like glass if the distances were regular and the waves just the right length.

Those waves would also be delayed, just like light in glass, air, or water. Interestingly, even light still moves through these mediums at light-speed, but all it's energy moves slower. If you had a sensitive enough detector you could see heavily attenuated light that didn't slow down.