Could Rainsford turn out to be the next "General" on the island? Why or why not?

No, I don't believe that Rainsford will become the next General on the island. His views of hunting differed greatly from Zaroff's, and he would never continue the "game". He might hunt on the island, but it wouldn't be humans, it would be animals. It is doubtful that Rainsford would even remain on the island for long if rescued, but we have to remember, it might be awhile before a ship would get close enough to make rescue possible.