Sorry. I was stupid. You are right. On dimension 1, the quotient is smooth, it's certainly normal. Then I should ask for an example for 2).

Yes, I think you are right, the sheaf is the ordinary coherent sheaf flat over $Y$. What I am worried is the that the function $h^i$ is defined for all geometric points of $Y$ not points of $Y$. This is different from the statements in Hartshorne.

Thanks for the answer. This is exactly the thing I am looking for. Now I understand $\hat{A}$ is normal, and as a result it is a product of integrally closed domains. Then could you please explain why $\hat{A}$ is a domain provided that the closed subscheme Spec$(A/I)$ is connected? How is Spec$(A/I)$ related to the set of closed points in Spec$(\hat{A})$?

Thanks for the explanation. I proved that in this case, $\hat{R}_I$ is an integral domain by showing that it is subring of the integral domain $k[y][[x]]$. But yours is a much simpler proof.

Yes, thank you. So $f'(a_0)$ means $f$ is etale at that point $a_0$. This is exactly what I was looking for.