In the US you have to win an election if other people in your party want to run for the office before you can run for the real election. But sometimes the party just names a nominee if there's no objections.
Very possible if he does a good job. Tennessee isn't a hotbed of politics, so his celebrity would be a big deal there, as the state has always been a top wrestling business territory. But that would be years from now.
Like Jerry Lawler ran for mayor of Memphis twice, but his questionable personal life stopped him from winning even though he's one of the biggest celebrities in the city. Kane has no skeletons in his closet, he's a really good person from everything anyone has ever said.