What is an adjective to describe the duality of a first responder (such as an EMT responding to a terrorist attack) who is terrified for the wounded, yet simultaneously calm, detached, and capable enough to give aid?
The following don't explicitly satisfy the terrified criteria; they are instead words that represent calmness under pressure. Terror could be construed to be a form of pressure.
Equanimity is defined as:
evenness of mind especially under stress
Its adjective would be equanimous.
Its synonym, sangfroid, is defined as:
self-possession or imperturbability especially under strain
While the dictionary does not note its adjectival properties, you can probably use it as one. A sangfroid calm is a reasonably common phrase.
Unflappable is similarly defined:
having or showing calmness in a crisis
Someone commented on the OP's question with steely and I think that best fits the situation and usage that you describe.
steel·y adjective /ˈstēlē/ steelier, comparative; steeliest, superlative Resembling steel in color, brightness, or strength - a steely blue Coldly determined; hard - there was a steely edge to his questions