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The answer to Question 1 is negative. Let $G=\{\{1, \dots, N+1\}\}$ and $H$ consist of all subsets of $\{1, \dots, N+1\}$ of size $N$. If $K$ is a distinguisher for $G$ and $H$, then for each $i \in \{1, \dots, N+1\}$ there must be a set $A \in K$ such that $A \cap \{1, \dots, N+1\}=\{i\}$. Thus, $|A| \geq N+1$.