With two exceptions, a goal keeper may deliberately handle the ball within their own penalty area.
No. Only in the penalty area may the goal keeper handle the ball.
As long as the ball is inside their penalty area a goal keeper may handle the ball. The position of the ball is important. The position of the goal keeper is not.
A goal keeper cannot deliberately handle the ball outside of his own penalty area. If they do, then a direct free kick is awarded to the opponents.
In short, no. A goal kick is not complete (the ball in not yet in play) until the whole of the ball crosses the whole of the penalty area line and into the field of play. If the goalkeeper touches the ball before it completely leaves the penalty area, the goal kick must be retaken. If the goalkeeper touches the ball with his hands after the ball has left the penalty area and become "in-play", then he is guilty of a handling infraction because the ball is not within the keeper's own penalty area anymore. Either the ball is still in the penalty area or it isn't; it can't be both. This is all without even considering the fact that a goalkeeper cannot handle a ball that has been intentionally kicked to him by a teammate, the so-called "pass-back" rule.
If their team mate passed the ball deliberately, with their feet, and the goal keeper then touches it with their hands within their own penalty area, then an indirect free kick is awarded to the opponents at the place where the goal keeper handled it.
No because it will be classed as handball and the other team will get a free kick.
This depends on the referee's determination of the passer's intent. If the player was deliberately passing the ball to the keeper, then the keeper may not handle the ball without penalty (though of course saving it with the feet or another part of the body is perfectly legal).
A goal keeper may not handle the ball if it is passed to them from the feet of a team-mate. If they do, an indirect free kick is awarded to the opponents at the location of the handling.
The purpose of the penalty arc is to ensure that, when a player takes a penalty kick, his teammates and opponents stay ten yards away from him at all times. While the goalkeeper has the ball in his possession, no player is allowed within the penalty arc until the keeper releases the ball.
If a defender, one not the goal keeper, deliberately handles the ball within their own penalty area then the restart is a penalty kick for the opposing team. It could also be misconduct depending on the circumstances. If an attacker deliberately handles the ball within the opponent's penalty area, the restart is a direct free kick for the defense. It must be, in the opinion of the referee, deliberate.
The goal keeper (may be known as a goalie or keeper) is allowed to touch the ball with his hands and to pick up the ball as long as he is within the 18 yard box (also known as the penalty area). Other players may only handle the ball when it is out of play. This is usually either when the ball goes outside the bounds of the field, such as for a throw-in, goal kick or corner or when the referee has blown his whistle to indicate a foul or to indicate the end of a half. Note that during a goal kick, corner or free kick / penalty the player may initially position the ball with his hands, however once the referee blows his whistle to recommence play, the ball must only be played with the feet.