Unless $Y$ is generic in your $X$, then what you describe under #3 is about the only case this will be true. Here is why:

 - *normal* is equivalent to $S_2$ and $R_1$
 - *reduced* is equivalent to $S_1$ and $R_0$

So, if you take a regular hypersurface in something normal, you reduce both the $S$ and the $R$ properties by one, so you win. However, as soon as you take something codimension $2$, you might end up in the singular locus of $X$. 

For a concrete example, take $X=Z(xy-z^2)$ (or just about anything singular, but normal) and take $Y=X\cap Z(x,y)$, which is a fat point at the origin.

On the other hand, if $Y$ is a general complete intersection in $X$ and $X$ is reduced, then so is $Y$, because both being $S_1$ and $R_0$ are inherited by general hyperplane sections.