Why didn't Rainsford spare Zaroff's life?
I think that there are at least two possible answers. You can pick which one you like better based on your reading of what kind of a man Rainsford is.
The first possible reason is that Rainsford is sort of executing Zaroff for all his crimes. Zaroff has confessed that he is, in essence, a mass murderer. Since he has, you can argue that he deserves to die and that Rainsford is just taking justice into his own hands.
A second possible reason is that Rainsford wants to take over Zaroff's hunting operation. Remember that he is a great hunter and might share some of Zaroff's attitudes. He might have realized that hunting people would be a lot more fun than hunting animals. Maybe that is why he finds Zaroff's bed so comfortable -- maybe that's a metaphor for him wanting to step in and take Zaroff's place as the hunter.
Rainsford is the true hunter. He certainly did not like being on the side of the prey, but he can't escape his true identity. I think after he learned what Zaroff did, he wanted Zaroff to die whether by his hand or another way.
Likewise, Rainsford was still playing the game. The game was essentially a fight to the death between two men. Rainsford had legitimately outsmarted Zaroff for 3 days, although on the first day Zaroff had extended him grace. That extention just served to further the complications of the rising action. If Rainsford did not kill Zaroff, I do not honestly believe that Zaroff would have let him off the island alive to go and tell the authorities what Zaroff does. Zaroff is smarter than that. The way I see it, Rainsford had no choice.