Is this operator positive? When is the operator $\begin{pmatrix} S^*S & \sqrt{2} S \\ \sqrt{2}S^* & S^*S \end{pmatrix}$ positive, where $S \in B(H)$ is of norm $\sqrt{2}$.

It is not true in general. To see this, set
$$
S=\left(\begin{array}{cc}
0 & 0\\
\sqrt{2} & 0
\end{array}\right).
$$
Then
$$
S^{\ast}S=\left(\begin{array}{cc}
0 & \sqrt{2}\\
0 & 0
\end{array}\right)\left(\begin{array}{cc}
0 & 0\\
\sqrt{2} & 0
\end{array}\right)=\left(\begin{array}{cc}
2 & 0\\
0 & 0
\end{array}\right),
$$
so that your matrix takes the form
$$
A=\left(\begin{array}{cccc}
2 & 0 & 0 & 0\\
0 & 0 & 2 & 0\\
0 & 2 & 2 & 0\\
0 & 0 & 0 & 0
\end{array}\right).
$$
Note that the "middle" submatrix is
$$
B=\left(\begin{array}{cc}
0 & 2\\
2 & 2
\end{array}\right)
$$
which has determinant $0\cdot2-2\cdot2=-4<0$, so that $B$ is not
positive semidefinite. Thus, $A$ is not positive semidefinite either.
