No. Roughly speaking, finite group schemes are objects with slope between 0 and 1 and so there is no Poincare duality without twisting except on curves (see the article of Artin and Milne for that). There is a flat analogue of the etale $\mathbb{Z}_l(r)$ duality --- see the article of Milne you mention and later articles. I think the affine Lefschetz theorem fails already for $\alpha_p$.
anon
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