We consider properly immersed two-sided stable minimal hypersurfaces of dimension . We illustrate the validity of curvature estimates for (and associated Bernstein-type properties with an extrinsic area growth assumption). For we illustrate sheeting results around "flat points". The proof relies on PDE analysis. The results extend respectively the Schoen-Simon-Yau estimates (obtained for ) and the Schoen-Simon sheeting theorem (valid for embeddings).