Yes. In fact, one of the Momofuku recipes involve shaving frozen foie gras.

It will freeze well although it can get a little more mushy then fresh after thawing.

Sure thing. In fact, it's often sold frozen.

My mother just today brought a pound and a half of foie gras to store in my freezer. She bought it from some friends who import all sort of fancies and it came frozen so my guess is a hopeful YES.