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, $|K| \geq N+1$.