You asked more than your entitled one question, so I have had to edit your question down to focus on one aspect only of this excellent short story - mood. When we think of mood in literary terms, we refer to how the story makes us feel when we read it. Clearly to work out the mood we need therefore to be aware of what impact the author is trying to have on us.
The last part of this story, therefore, which features the enacting of "The Most Dangerous Game", is designed to create a mood of great suspense in us, the readers, as we, from the point of view of limited third person perspective, see the game enacted from Rainsford's perspective. We see how he tries to trick Zaroff and how he fails, only to ultimately win. Passages such as the following help create this mood:
Rainsford held his breath. The general's eyes had left the ground and were travelling inch by inch up the tree. Rainsford froze there, every muscle tensed for a spring. But the sharp eyes of the hunter stopped before they reached the limb where Rainsford lay; a smile spread over his brown face. Very deliberately he blew a smoke ring into the air; then he turned his back on the tree and walked carelessly away, back along the trail he had come. The swish of the underbrush against his hunting boots grew fainter and fainter.
Here we see Rainsford waiting to trigger his trap and launch himself upon Zaroff, but just before he does so, Zaroff seems to realise what he is trying to do and retires.
Passages like this one clearly establish the mood of suspense. As Rainsford draws his breath, so the reader does to, as he waits to see what will happen.