Counter Example for Going Down Theorem
Note that $R\simeq\mathbb R[t^2-1,t^3-t,z]$ by $x\mapsto t^2-1$, $y\mapsto t^3-t$, $z\mapsto z$, and $S=\mathbb R[t,z]$.
Now I change the notation and set $S=k[t]$, and $R=k[t^2-1,t^3-t]$. (Note that $R=\{f\in k[t]:f(-1)=f(1)\}$.) Moreover, $S[z]$ is the integral closure of $R[z]$ since $S$ is the integral closure of $R$. To show that Going-Down fails for $R[z]\subset S[z]$ consider the prime ideals $\mathfrak p=(z-t-1)\cap R[z]$ and $Q=(t-1,z)\subset S[z]$.
We show that $Q$ lies over $\mathfrak q=(t^2-1,t^3-t,z)\supseteq\mathfrak p$, but $S[z]$ has no prime in $Q$ lying over $\mathfrak p$.
Observe that we also have $(t+1,z)\cap R[z]=\mathfrak q$, so $\mathfrak p\subseteq\mathfrak q$. Since $z\in\mathfrak q\setminus\mathfrak p$ the inclusion is strict. Suppose there is $P\subset Q$ lying over $\mathfrak p$. Then $P$ is principal generated by an irreducible polynomial, say $f$, and $f(1,0)=0$. We have $(f)\cap R[z]=(z-t-1)\cap R[z]$. Since $(t^2-1)(z-t-1)\in(z-t-1)\cap R[z]$ it follows that $(t^2-1)(z-t-1)\in(f)$, so $f\mid(t^2-1)(z-t-1)$. From $f(1,0)=0$ we get the only possibility $f=t-1$. Now $(t^2-1)(t-1)\in(f)\cap R[z]$, so $(t^2-1)(t-1)\in(z-t-1)$, a contradiction.
I followed the sprit of B. Heinzer, http://www.math.purdue.edu/~heinzer/teaching/math557/gdown.pdf.
Let $k$ be an (algebraically closed) field of characteristic zero, and $R = k[x,y,z]/(y^2-x^2-x^3)$. We are going to show that the going down property fails between $R$ and $S$ its integral closure of $R$.
Let $K = Q(R)$ be the field of fractions of $R$. Since $\frac{y}{x} \in K$ satisfies the equation $T^2 - 1 - x = 0$, $R[\frac{y}{x}] \subseteq S$. Notice that $R[ \frac{y}{x} ] \cong k[x,y,z,U]/ (xU- y, U^2 -1 -x) \cong k[z,U]$. Hence $S = R[\frac{y}{x}]$.
We compute $J$ the Jacobian ideal of $R$: $J = I_1 ([ -2x -3x^2 \quad 2y \quad 0]) = (2x+3x^2 , 2y)R = ( 2x+3x^2, y)R$. Since $R$ is a complete intersection, the non-normal locus is determined by $J$.
Let $Q = (\frac{y}{x} - z)S$. Then $y-xz , z^2 -1 -x \in Q \cap R$. If $J \subseteq Q \cap R$, then $y \in Q \cap R$. Then the height of $Q \cap R$ is at least $2$, and this is a contradiction. Therefore, $p = Q \cap R$ does not contain $J$. Hence $R_p$ is a DVR (so it is integrally closed), and we have $S \subseteq R_p = S_p$, i.e., $Q$ is the only prime lying over $p$. In fact, $p = (y-xz, z^2 - 1-x)R$. (Check that $R/ (y-xz, z^2-1-x)$ is isomorphic to $k[z]$.)
We are going to construct a maximal ideal $M$ of $S$ such that $p \in M$, but $Q \nsubseteq M$. Let $M = (x,y,z+1, \frac{y}{x}-1)$. It is easy to show that $Q \nsubseteq M$, but $p \subseteq M$. Notice that $M \cap R$ is a maximal ideal, so $p_1= M \cap R \supsetneq p$. But there is no prime in $M$ which contracts to $p$. (Recall that $Q$ is the only prime lying over $p$, and $Q \nsubseteq M$.) Hence the going-down property fails.
The argument can to modified to work with non algebraically closed field $k$. The only place we use the assumption is using the Jacobian ideal to say $R_p$ is a DVR. This can be easily checked: Since $x$ are not in $p$, $R_p = R[\frac{1}{x}]_p$. But in $R[\frac{1}{x}]$, $p R[\frac{1}{x}] = (y-xz, z^2 -1 -x)R[\frac{1}{x}] = (\frac{y}{x}-z, z^2 - 1 -x) R[\frac{1}{x}] = (\frac{y}{x} - z)R[\frac{1}{x}]$.
I would like comment that an assumption on the characteristic of $k$ is necessary ($2 \neq 0$) in order to show the (non)-containment $Q \nsubseteq M$.