Handle the ball in the goal area only
No, the only place a goalkeeper may handle the ball is in his own penalty area.
The goalkeeper is permitted to handle a ball received directly from an opponent's throw-in. If the throw is from a teammate (or himself), an indirect free kick will be awarded to the other team.
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.
No, he can't
A goalkeeper may not handle a ball that has been deliberately kicked to him from a teammate. If the goalkeeper does handle the ball after is was deliberately kicked to him from a teammate, the opposing team is awarded an indirect free kick from the place where the ball was at the moment it was touched (or on the goal area line parallel to the goal line nearest to the spot of the ball if it is within the goal area).
Certainly he can. The penalty area allows him to handle the ball WITHIN that area, but as long as the tackle is legal, he can be challenged wherever he is on the pitch.
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.
The penalty area.
The area within which a goalkeeper may usually legally handle the ball is called the penalty area, and measures 44 yards across and 18 yards deep. The goalkeeper may leave this area, but is treated like any other player while outside of the penalty area.
The so-called "Back Pass" rule means that the goalkeeper cannot handle the ball, even within his own penalty area, if it was deliberately kicked to him by a teammate. If the goalkeeper violates this rule, an indirect free kick is awarded to the attacking team at the point where the goalkeeper handled the ball. Note that a penalty kick can never be awarded for a goalkeeper's handling.
It is called the penalty box and is the biggest of the two boxes and is touching the half circle or penalty arc.