All the standard counterexamples to flatness that I have seen involve completions with respect to non-finitely generated ideals. I am interested in the following two cases:

Let $A$ be a local ring with finitely generated maximal ideal $\mathfrak{m}$. Is the completion morphism $$ A \rightarrow \hat{A} := \underset{n \in \mathbb{N}}{\mathrm{lim}} \text{ } \frac{A}{\mathfrak{m}^{n+1}} $$ conjectured to be injective? In other words, is Krull's intersection theorem expected to hold?

Let $A$ be a ring and let $I$ be a finitely generated ideal of $A$ such that $A/I$ is Noetherian. Is the completion with respect to $I$ expected to be flat?

Thank you in advance!