In football, is a goalkeeper allowed to pick up the ball and place it back down onto the penalty mark before the referee starts the penalty kick? I do not see any rules in FIFA that would permit or forbid this action, so I was wondering if this would be legal.
You won't find any law against it because it is completely legal.
It is a bit of a stretch see how this could be a problem. Any controversy here would most likely arise from the goalkeeper placing the ball on an uneven section of turf in order to impair the kicker. If this was to happen, the kicker is then within their rights to place the ball however they like (ie. on a flatter part of the penalty mark), provided it is still on the penalty mark and any readjustment is done in a timely manner.
What would be illegal here, would be the goalkeeper delaying the restart of play or dissenting the decision by throwing the ball away, refusing to return to his goal line for the kick to be taken, or entering into protracted disputes over the placement of the ball.
If this occurred, the referee would either verbally warn, or caution the goalkeeper for dissent or delaying the restart of play.