Big words or what? What are you trying to explain?
After hearing the raucous, cheering response to Adam’s speech, Mr. James feared that his new student body president might be an amateur demagogue.
A participle has the form of verb + -ing. So the only such word here is 'smiling' that is the participle in this sentence. It modifies 'woman', that is, it refers to how the woman opened the door - smiling pleasantly.
It would be the first sentence. chronological means from first to last and the first one would be the only one that makes sense.
