The object of a verb is usually marked with を, but when that object is the topic of the sentence, を will be replaced with は. This is a basic grammatical rule, and it has nothing to do with the usage of 通して itself. See: What is a topic prominent language?
In your example, みなさんのパフォーマンス is the topic of the sentence, so it's marked with は even though it's the ...
I believe this usage corresponds to the following definition of 通す/徹す from 大辞林:
So ~を通して聞く would basically be "listen to 〜 straight through, from beginning to end."
いきなり is an adverb meaning "out of nowhere", "all of the sudden". Forget "without warning" for now. It plainly modifies 言い出す as an adverb. 言い出す is the first verb after いきなり, so it cannot be simpler.
何を is "what". 何 is the object of 言い出す.
言い出す is "to start saying", "to bring up (a topic)". Its subject is the girl.
か is the question marker.
と思えば is the plain ...