small size and small depth circuit for set intersections
You are asking many different questions. The usual rule is one question per post. To help you get started, here is how to compute $S_1 \cap S_{\min S_1}$ (defined as $\emptyset$ if $S_1 = \emptyset$).

The input to the circuit is the $n^2$ variables $x_{i,j}$, indicating $i \in S_j$ (this means that they are TRUE if the condition holds, and FALSE if it doesn't hold). The output to the circuit are $n$ variables $y_i$, indicating whether $i \in S_1 \cap S_{\min A}$.

Let $f_a$ indicate $\min S_1 = a$. Then
$$
f_a = \lnot x_{1,1} \land \cdots \land \lnot x_{a-1,1} \land x_a.
$$
Given these, we can calculate
$$
y_i = x_{i,1} \land \bigvee_{a=1}^n (f_a \land x_{i,a}).
$$
We can implement each $f_a$ as a depth 2 circuit with unbounded fan in of size $O(a)$, and each $y_i$ as a depth 2 circuit (with inputs $f_a$ and $x_{i,j}$) with unbounded fan in of size $O(n)$. The circuits for $f_a$ take $O(n^2)$ in total and the circuits for $y_i$ take $O(n^2)$ in total, so altogether we get a depth 4 circuit of size $O(n^2)$.

This circuit is equivalent to a depth $O(\log n)$ circuit of size $O(n^2)$ with bounded fan-in, by replacing each large fan-in gate by a balanced binary tree.