Take $X=Z(xy-z^2)$ (i.e., a quadric cone, which is normal). The element $x$ is regular (e.g., since $X$ is integral) and let $Y=X\cap Z(x)=Z(x,xy-z^2)=Z(x,z^2)$ which is a double line and hence not reduced. So, even for your claim in #3 you need to assume more, say something that assures that $Y$ is not entirely singular. For instance you may assume that it is a general hyperplance section.
What you have is that
- normal is equivalent to $S_2$ and $R_1$
- reduced is equivalent to $S_1$ and $R_0$
So, if $Y$ is a regular hypersurface in something normal and it is not entirely singular (i.e., it is $R_0$), then it is reduced. With higher codimensional subschemes the problem could be that $Y$ may very well end up inside the singular set of $X$
At the end the point is that you have to ensure that $Y$ is not entirely singular and I don't see a good way doing this other than assuming that it is cut out by general hyperplane sections unless you have some specific information. Once you have that, then you need that $X$ is at least $S_{t+1}$ where $t=\mathrm{codim}_X Y$.