I believe that Helicon Hall is the same building mentioned here (https://en.wikipedia.org/wiki/Helicon_Home_Colony) which burned in 1906. I'm guessing that people died in the fire because there were no fire escapes (hence, they were missing). When the young man says, "Do I look like I'd climed (climbed) down..." it means, "Do you think I'm stupid/gullible/foolish?" Sorry, I googled "Persian dates" and came up empty-handed. Basically, the man with the money is trying to be kind and generous like other people he's heard about.

O. Henry was writing in 1910, humorously, to be entertaining. He makes contemporary references, uses slang, and enjoys saying simple things in complicated ways. This is not easy English, and some things in his stories are hard for native English speakers to understand. The Cardiff Giant was a hoax, a faked "petrified man" ten feet tall, exhibited by P. T. Barnum, that fooled people. He's making fun of the rich man's vague memory by throwing in something incongruous. He's trying to remember the legend of Caliph Haroun al-Rashid. In his stories, for some reason, O. Henry often makes many references to a story collection called "The Arabian Nights," ancient Persia, Caliph Haroun al-Rashid, the Rubáiyát of Omar Khayyam, and so forth. You'll notice that he introduces this story with a reference to New York as being "Bagdad-on-the-Subway" and gives things away with a reference to "Mr. H. A. Rashid." He is disguising the name of Caliph Haroun al-Rashid by writing it as if it were the name someone living in New York in 1910! He's analogizing 1910 New York to the imagined glory of the days of the Arabian Nights. Caliph Haroun al-Rashid is often portrayed as a wise and humane leader, and--as with other wise leaders--the story is told that he liked to disguise himself by dressing as an ordinary citizen, and go around Bagdad to talk directly to ordinary people. "Making Persian dates" is a pun. To "make a date" is to make an appointment to meet someone. But "a date" is also the fruit of the date palm tree and is also associated with the Middle East, so a "Persian date" also means the fruit.

For "Helicon Hall" I have to do an Internet search, and you should be prepared to do them if you're going to try to understand everything in O. Henry. You don't need to understand everything to understand the story, though. I learn that Helicon was a failed Utopian community, founded by Upton Sinclair (don't worry about who he was), and the main building burned down in 1907. It had no fire escapes, and there was controversy about whether it should have had them; accusations that it was arson, that it was deliberately burned down by Sinclair's political opponents, and so on. James Turner speaks colloquial speech, using slang. He thinks Crowley is trying to swindle him. He means "do I look like somebody who is gullible and easy to swindle." He might have said "Do I look green to you?" or "Do I look like I'm just off the boat?" I don't know why he uses the phrase he does. But it is clearly just something that had been making news lately. This story is just FULL of things like this, and probably not the best O. Henry story you could have chosen. 90% of the fun is in O. Henry's choice of irreverent language. His language is a little like that of a stand-up comic. The point of the story, of course, is that Crowley wants to give Turner a fortune, and Turner rejects it because he doesn't believe it--he's too "streetwise"--and also because he knows what he wants, and as long as he has something to read and a place to cool his sore feet, he doesn't want to be bothered.