Two shot penalty, he must replace his ball and then you must play your own.
Yes, one stroke penalty, and you must play the ball as it lies. If it was however stopped or deflected by an opponent or their equipment, the ball must be played as it lies and there is no penalty.
The 3 yard line.
No, but you will risk losing the ball you are playing with and most likely someone will take it for their own.
It really depends on the penalty and wherethe ball carrier is. If ball would be spoted where the penalty occured, yes. If the ball carrier didn't make it out of the end zone, it is a touch back and the penalty yards will be assessed from the 20 yd line. If he did make it out, the penalty yards will be assessed from he was downed or, he scores a touchdown they will be assessed on the ensuing kickoff. Yes. A penalty committed by a team in possession of the ball in its own end zone results in a safety.
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.
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.
Yes so long as he did not receive the ball from one of his own team mates.
A foul does not have to be near the ball. A foul must occur (1) by a player, (2) on the field of play, (3) against an opponent, and (4) while the ball is in play. If the foul you describe is by a defender, against an attacker, in the defender's penalty area then play is stopped and a penalty kick awarded. Depending on the severity, it could also be misconduct.
His own penalty area? Assuming it is not the goal keeper, it is a penalty kick to the opposing side, he may be given a yellow or red card. In the opposing penalty area it is a free kick to the opposing side and usually a yellow card.
A goal keeper may never use their hands outside of their own penalty area. A goal keeper may not use their hands, even in their penalty area, when a team-mate deliberately plays the ball to them with their feet. A goal keeper may not use their hands, even in their penalty area, when a team-mate directly throws the ball to them on a throw-in.
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.