Keeper Handling?
 

Is it illegal for a keeper to leave the penalty area, kick a ball that was last played by an opponent into the penalty area and then pick it up?

no. it would only be a violation if a teammate had kicked the ball to the keeper (passback misnomer). a keeper is equal to any player and may play the ball the same as any player anywhere on the fop. the keeper's only advantage is the use of hands within his/her own pa. the prohibition on using hands by the keeper within his/her own pa only applies if the ball was last kicked to the keeper by a teammate, thrown-in by a teammate, held and released and rehandled by the keeper, or held for more than 6 seconds.