Is the ball alive or dead when hitting a runner in fair territory?

It depends. Is the ball being thrown or is it hit. If it's hit off the bat and hits a base runner it's a dead ball and the runner is out if the ball has not passed a fielder. If the ball has already passed a fielder then the ball is live and the runner is not out. It is as if it never happened. If it hits the runner when it is thrown it is perceived as if it never hit the runner.