Can an opposing player take the ball from the goalie, inside the goalie's area, when the goalie has placed it on the ground to kick it? The ball is no longer in the goalie's hands at this point, but rather, has been placed on the ground. Which rule in the FIFA rules explains this best?
If it's in open play, this is perfectly fine as long as it's not a foul against the goalie and the goalie does not touch the ball with his hand. If the goalie put down the ball to do a free kick or a goal kick, no opponent is allowed to touch the ball as it is not open. See the 9th law of the game.
Here you'll find a few (and related) examples of the situation you are describing.