1. Is any family of hypersurfaces of fixed degree d over a non-reduced base flat? This is true over reduced 
bases (assume everything Noetherian), since the Hilbert polynomial is constant. Is there any criterion of flatness for families over non-reduced schemes?

2. Is there any result that guarantees that push forwards and higher push-forwards (via a proper mophism of Noetherian schemes) of a flat over the base sheaf are locally free if the base is non-reduced?

Thanks a lot!