I've been faced with the following sentence:
What I'm wondering about it is the "聞かれて" part.
I know that Kikareru would be the passive form of Kiku - to listen.
So, what is this ~te ending doing there?
The same thing any te form does. It's "continuative" and the part that comes after elaborates on that condition. So if you say あんな事聞かれて・・・ then whatever comes after will be in the context of having been asked such a question. So 聞かれて平常心でいられるか means that, having been asked such a thing, the speaker wonders if he/she/whoever is being asked will be able to stay calm.