I don't think *normal* is quite enough, but something similar should work. So, 

 - *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 which follows from you genericly reduced assumptiom, then $Y$ is reduced.
(This is just restating what you are saying in #3).

So, a simple sufficient condition is to assume that $X$ is at least $S_{t+1}$ where $t=\mathrm{codim}_X Y$. That way $Y$ will be $S_1$ and your generically reduced assumption implies that it is $R_0$. 

Here is an example that the question in #3 only holds if the codimension of $Y$ is $1$:

Let $X = \mathrm{Spec} k[x,y,z,t,w]/(xz,xt,yz,yt)$. This is a threefold in $\mathbb A^5$. Near any point with any of $x,y,z,t$ non-zero, this is locally isomorphic to $\mathbb A^3$, hence smooth. When $x=y=z=t=0$, that's a line in $\mathbb A^5$, so $X$ is smooth away from that line, so it is $R_1$. We will see this below that it is $S_2$ and hence normal.

Next let $Z = Z(w)\subset X$. This is the union of two planes meeting in a single point, the famous example of something not normal, not Cohen-Macaulay, etc.. It is easily seen to be $S_1$: For instance $x+z$ is a regular element, but modding out by that we get $Y=\mathrm{Spec} k[x,y,z,t]/(x^2,xy,xt,yt)$ which is two lines intersecting in a fat point, which is $R_0$, but not reduced at the fat point and hence non-reduced and cannot be $S_1$. 

So, 

 - $Y$ is $S_0$ and $R_0$, in particular non-reduced, 
 - $Z$ is $S_1$ and $R_1$, in particular reduced, but not normal
 - $X$ is $S_2$ and $R_1$, in particular normal.

So, $Y$ is the complete intersection of a regular sequence, $w, x+z$ in the normal $X$, it is generically reduced, but not everywhere reduced.

I think you can adapt this example to show that if $X$ is not $S_{t+1}$, then there is a codimension $t$ complete intersection which is generically reduced but not everywhere reduced.

Otherwise there is still taking *general* hypersurface sections, those retain even normality.