It depends.
In High School the batter/runner is safe if he touches the base before the ball is caught. BEFORE is the key word here. If it is a tie, the runner is not there BEFORE the ball and is therefore out. Pro rule states that the batter/runner is out if the ball arrives BEFORE he gets to the bag. So therefore if it's a tie in the pros, the runner is safe.
In your second scenario it should have been a no-pitch, a do-over.