I want to make it clear whether we can use the same measures of executive functions such as working memory under neutral and emotionally charged situation and claim it as cool and hot executive functions?
Seems feasible as long as the hot measure is emotionally and motivationally significant, whereas the cool measure is emotionally neutral. Succeeding on the hot measure should be facilitated by regulating emotions.
I would say that this should not be done when it is the same executive function, though you might speak of "hot" vs. "cool" conditions. But I very much like the distinction you are drawing! And it seems possible that there are highly similar, but nonetheless distinct, executive-type mechanisms used for emotionally charged (vs. relatively neutral) situations.
No. Although I agree that operational definitions of hot and cool are not always fully clear, in general the term "hot" executive functions refers to the control of emotionally loaded situations. The "cool" measures, instead, involve tasks and materials that are emotionally neutral, where the conflict is purely cognitive (e.g., between a prepotent and a less dominant response, or between salient, irrelevant and less salient but relevant information). Regarding the more general problem of definitions and measures of executive function, i.e. "which tasks measure what", please see Article Which tasks measure what? Reflections on executive function ...