What are some figures of speech in "The Green Door" by O. Henry?

1) PERSONIFICATION (speaking about non-human entities as if they were human):

a) "In the big city the twin spirits Romance and Adventure are always abroad seeking worthy wooers. As we roam the streets they slyly peep at us and challenge us in twenty different guises."

"Romance" and "Adventure" are not people who can actually "roam the streets" and "peep at us" and "challenge us."

2) ALLITERATION (the repetition of consonant sounds at the beginnings of words, or in accented syllables):

a) Usually he passed the dispenser of the dentist's cards without reducing his store; but tonight the African slipped one into his hand so deftly that he retained it there smiling a little at the successful feat.

3) SIMILE (a comparison that uses the word "like," "as," "similar to," or other words of comparison):

a) Every half minute he chanted a harsh, unintelligible phrase akin to the jabber of car conductors and grand opera.

The word "akin" means "like," or "similar". The author is comparing the man's speach to the "jabber of car conductors and grand opera"; in other words, it is unintelligible.

4) OXYMORON (The use of contradictory terms side by side):

a) She began to eat with a sort of dainty ferocity...

"Dainty" and "ferocity" are two opposites, yet the author has combined them in one phrase.

5) SYNECHDOCHE (when a part is used to represent a whole, or a whole to represent a part):

a) She began to tell him her little story. It was one of a thousand such as the city yawns at every day.

A city cannot yawn.  Rather, the author means that the inhabitants of the city tend to yawn when they stories like the girl's.  Thus, the individual parts (the inhabitants) are represented by the whole (the city).

This passage could also be considered personification: the author speaks of the city as if it were a person who can yawn. 

What examples of hyperbole and paradox are in O. Henry's "The Green Door"?

In his story, "The Green Door," O. Henry employs hyperbole, or obvious exaggeration, for humorous effect.  In defining true adventurers, O. Henry writes that the Prodigal Son is a "fine example--when he started back home."  Then, he describes Rudolf Steiner as a "true adventurer," explaining that there are few evenings that he did not go forth from his "hall bedchamber in search of the unexpeted and the egregious."  However, since Rudolf simply looks for adventure in his immediate area, his actions seem somewhat less than daring.

Paradox is an apparent contradiction--a statement that, while appearing to be contradictory reveals a kind of truth.  Like other writers, it seems that O. Henry uses paradox to express the complexity of life by showing how opposing ideas can be both contradictory and true.  Rudolf Steiner, "the adventurous piano salesman," enters the building where he conceives that "his adventure must lie" and discovers behind the green door a starving young shopgirl.  After becoming enamored of her, Rudolf fetches food and hot tea for her; they dine together and, the Romantic that he is, Rudolf becomes jealous at the idea that someone else might have come to her door.  Also, he lies about why he has come to her door so that she will not know about the card that has directed him to her"strange expedient."  As he shuts the door, ironically, Rudolf notices that there are several green doors, and then learns that the Negro has been handing out cards advertising a play in town that is called, "The Green Door."

While Rudolf is not adventurous enough to learn what the play is about or to check the entrances behind the other green doors, he is certainly romantic enough to believe that is was Fate "that doped out the way for me to find her."  So, the truth of the paradox, or apparent contradiction, is that Rudolf is adventurous enough to embark upon a romance with the pretty little shopgirl.