It is easy to see that (1) holds, since we can detect regularity by looking at the dimensions of the Zariski tangent spaces at the points of

$$\mathscr{Y}\times_{R}R/(t^{2})\cong\mathscr{Z}\times_{R}R/(t^{2}).$$

Moreover, the special fibers of $\mathscr{Y}$ and $\mathscr{Z}$ are isomorphic so that $\mathscr{Z}_{k}$ is a divisor with strict normal crossings. Point (2) is more subtle and follows from [Ki03, 2.6(2)]. ∎