Do objects absorb or reflect gravitational waves?

Yes, objects can scatter and absorb gravitational waves.

For example, this paper considers the scattering and absorption of gravitational plane waves by rotating black holes.

And this answer discusses how much energy the Earth absorbs when a gravitational wave passes through it. (It’s miniscule.)

Historically, Feynman’s sticky bead argument helped settle early controversies over whether gravitational waves carried energy that could be absorbed and in principle detected.