It must be a momentum transfer problem and the graphene sponge extremely light. The height variation coming from beam divergence.
The question comes down to how good is their vacuum as to whether it is the momentum of reflected photons or air pressure on the hot side. The latter being the strong effect which powers toy radiometers.
The electrons ejected would leave the puck positively charged and attract it. If the thing works as described then it is either radiation pressure acting on a lightweight structure or more likely gas pressure depending on just how good their vacuum actually was.
Basically it is a very crude Crooke's radiometer with graphene sails.
If their laser power is extreme enough it could be surface ablation too.
Almost certainly. Separating electrical charge invariably creates an attraction between the positive and negatively charged components.