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 set 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.