Is the ball out of play if any part of the ball overlaps either the goal line or the touch line?

No, the whole of the ball must cross the line.

A player asks to leave the field of play and, as he is walking off, the ball comes towards him and he kicks the ball into the goal. What action does the referee take?

The player is cautioned. The game is restarted by an indirect free kick, taken by a player of the opposing team, from the place where the infringement occurred.*

A spectator blows a whistle and a defender inside his own penalty area picks up the ball, assuming that play has been stopped. What action does the referee take?

He awards a penalty kick.

The referee fails to see an assistant referee signal an offside offence and the ball enters the goal. Play is restarted with a kick-off and only then the referee sees the assistant referee's signal. What action does the referee take?

He allows play to continue since he cannot change the decision to award a goal after play has restarted.

When play is about to be started with a dropped ball the players of one team refuse to take part. What action does the referee take?

He merely restarts play by dropping the ball. It is not necessary for each team to be present for the restart with a dropped ball.

When the ball is dropped to restart play it bounces across the touch line without having been touched by a player. What action does the referee take?

He restarts play with a dropped ball at the same position as before.

The ball accidentally hits the referee on the field of play and rebounds into the goal. Is a goal scored?

Yes.

May a goalkeeper join other players at a dropped ball situation?

Yes. Any player may take part.