Is a parameter-independent support sufficient to belong to an exponential family? From Wikipedia (emphasis mine):

As a counterexample if these conditions are relaxed, the family of uniform distributions (either discrete or continuous, with either or both bounds unknown) has a sufficient statistic, namely the sample maximum, sample minimum, and sample size, but does not form an exponential family, as the domain varies with the parameters.

Does this mean that if the support (not domain) of a particular distribution does not vary with its parameters, then it is a member of an exponential family of distributions?

NO
A $t_{\nu}$ distribution has support on all of $\mathbb R$, no matter $\nu$.
However, $t_{\nu}$ does not have a moment-generating function, which is a requirement to be an exponential family.
