no because the keeper if has two hands on the ball you can not kick it but if the keeper has one hand on the ball you may kick it out of the keepers hands.
most of the time when a goal keeper takes the penalty shot but sometimes when the goal keeper of the other team is so much above his field or area the goal keeper may kick the ball from goal to the other goal.
It is legal for a teammate to kick the ball to their own goal keeper. The goalkeeper would not be allowed to use their hands. If they do so, then it would be an indirect free kick for the opponents at the location the goalkeeper touched the ball. A penalty kick would not be awarded.
If an opposition player pushes the goal keeper it is considered a foul and a direct free kick is awarded for the keeper's team. This is true for any player and not just the goal keeper.
yes, he can. for instance, when he tries to catch the ball which is coming from a corner kick, also it is not on target (not going for goal), the keeper accidentally punches the ball into his own goal. so it is counted as an "own goal" for the keeper.
Yes, but he cannot use his/her hands.
No, the keeper cannot pick the ball up after his own player intentionally plays it to him using the feet.
A goal keeper is allowed to touch the ball with their hands when the ball is inside their own penalty area. If a goal keeper does so outside of the area, then a direct free kick is awarded to the opposing team.
The goal keeper handling the ball outside of the penalty area is the same as any player deliberately handling the ball; a direct free kick. In the circumstance you describe the referee will see such minor violations as trifling and will warn the goal keeper to watch his line more carefully. Only flagrant or very frequent incidences will cause a direct free kick to be awarded. Often the keeper actually lets go of the ball in the area and makes foot contact with it outside the area.
If the ball crosses the goal line on it's own momentum, then a goal is scored, even if the keeper touched it. If it is a penalty kick during the match itself, then the kicker may only kick it again if it has touched another player, such as the goal keeper. If it is a penalty kick in extended time (e.g. time expired after the penalty kick had already been called) or if during kicks from the penalty mark (a.k.a. shootout) then no player may touch it a second time, except the goal keeper.