Yes, but he cannot use his/her hands.
I'm going to assume that you are asking about a ball that is deliberately kicked back to the goalkeeper by a teammate, then picked up. If the keeper touches the ball with his hands outside of the penalty area (the "18"), the offense is handling and the other team is awarded a direct kick from the spot where the keeper first touched the ball with his hands. If the keeper touches the ball with his hands inside the penalty area but outside the goal area (the "6"), the offense is goalkeeper handling and the other team is awarded an indirect kick from the spot where the keeper first touched the ball with his hands. If the keeper touches the ball with his hands inside the goal area, the other team is awarded an indirect free kick from the spot on the goal area line nearest where the ball was first touched by the keeper.
when the keeper or goalie is in possession of the ball after being received from a back pass or saving a shot
Yes. If the ball is kicked to them or thrown-in to them directly from a team-mate. If the goal keeper releases the ball from their hands they cannot pick it back up again until another player touches it.
No, the keeper cannot pick the ball up after his own player intentionally plays it to him using the feet.
They are usually called a goalie, or the keeper.
Yes, then he or she can handle it. If you just kick it back to him or her then they can't handle it and must kick it. If an opposing team member has made contact with the ball or had a shot, the goal keeper may handle the ball.
The ball is at your feet not hands
No.
They must allow the keeper room to kick the ball; 5-6 metres should be enough.
He can do so. The referee will stop play and restart play with a indirect free kick for the opposing team.
Because a shot with enough speed to cross the line and get past a keeper almost invariably does hit the back part of the net.