Let $X$ be an affine, irreducible (complex), generically reduced, scheme containing an embedded point, say at $x \in X$. Suppose further dimension of $X$ is strictly positive (can assume to be one dimensional). Assume that the associated reduced scheme $X_{\mbox{red}}$ is non-singular. Note that, there is a natural closed immersion $X_{\mbox{red}} \to X$. Then,

1) Are there examples of $X$ as above such that there exists a morphism $X \to X_{\mbox{red}}$ such that the composition $$ X_{\mbox{red}} \to X \to X_{\mbox{red}}$$ is the identity? I know of such examples in zero dimension. I am interested to see if such a phenomenon can happen in the setup of embedded points in schemes of positive dimension.

2) If such a map $X \to X_{\mbox{red}}$ exists, then does the structure sheaf of $X$ decompose (as an $\mathcal{O}_{X_{\mbox{red}}}$-module) as $\mathcal{O}_{X_{\mbox{red}}} \oplus I\mathcal{O}_{X_{\mbox{red}}}$ for some nilpotent ideal $I$?