Minimum number of intersections to arrive at specific set
Your problem (or rather, its decision version: are there $k$ sets whose intersection is a singleton) is NP-complete, so I doubt there is a simple solution that always works, even for your numbers.

The problem is clearly in NP. We show that it is NP-hard by reduction from SAT. Let $\varphi$ be a CNF on variables $x_1,\ldots,x_n$ and clauses $C_1,\ldots,C_m$. We form an instance of your problem on the universe $x_1,\ldots,x_n,C_1,\ldots,C_m,\Delta$. For every $x_i$ and every truth value $t$, there is a set $S_i^t$ which contains (1) $\{x_1,\ldots,x_n\} \setminus \{x_i\}$, (2) all clauses not satisfied by $x_i=t$, and (3) $\Delta$.

All sets contain $\Delta$. In order to get rid of all $x_i$, we need to choose at least one set $S_i^t$ for each $i$. The intersection consists only of $\Delta$ if and only if the choice corresponds to a satisfying assignment. Hence there are $n$ sets whose intersection is a singleton iff $\varphi$ is satisfiable.