For the first part of your question, *support* doesn't appear to be an appropriate word, as in the context of permutations it is normally used for the number of non-fixed points.

To define a term, let's define the process that arrives at the statistic:

Let $T_n$ be the set of all transpositions on a set of size $n$, say $S$.

Let $\pi$ be a permutation of $S$, and let $$\;_\pi T_n=\left\{\{t_1,\dots t_k\}\in T_n^k:\pi\prod_\limits{i=1}^k t_i=S\right\}$$

i.e. $\;_\pi T_n$ is the set of sets of transpositions that take $\pi$ to the identity $S$.

Define $\;_\pi U_n=\{u:u\in t,\;t\in \;_\pi T_n\}$, so that $\;_\pi U_n$ is the set of sets that contain the unique elements of a transposition set of $\;_\pi T_n$.

Define $\;_\pi V_n=\{|v|:v\in \;_\pi U_n\}$ and then the statistic we want is $\min(\;_\pi V_n)$.

So, **minimum cardinality of a identifying transposition set**.