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For #1, an example is given by two Moore spaces $M({\mathbb Z}/p^2,k)$ and $M({\mathbb Z}/p^3,k)$; the only cohomology is in degree $k$ and $k+1$ in characteristic $p$, and the ring and Steenrod module structures are trivial. This works with $p^2$ and $p^3$ replaced by any two powers of $p$ that are not $p$ itself (since for $p$ you get a Bockstein in the cohomology).