If your ball lands on the fairway or in bounds, there is no penalty for hitting a cartpath or any other sign. You have to play the ball where it lies just like if you hit a boulder or tree.
Fairway is the ground between the tee and the green from which the ball may be easily played.
No, the only time you are allowed to place the ball on a tee is when you are in the designated teeing ground hitting what is determined as the tee shot.
Yes you can move the ball a club length to the side
You can not ground your club (touch the ground behind the ball with the club head) in a fairway bunker. All bunkers are hazards either by the green or along the fairway.
Yes, as long as you do not disturb the ball.
Hitting a golf ball off the fairway results in it coming to rest in a less than desirable place. It could end up in a sand trap, a creek, or in the rough which is the higher cut grass adjacent to the fairway.
The only legal way of lifting the ball when it lies on the edge of the fairway is when the rules committee of a tournament is playing lift-clean- and place. This would be stated before the tournament began. This accounts for unsavory conditions such as rain. The player is allowed to place a tee where the ball was, clean the ball, and place it back in fairway.
You are entitled to a free drop if the ball is plugged in the fairway or closely mown area, that is apron, fringe etc, (nearest point of relief, no nearer the hole). If the ball is plugged in the rough, there is no relief.
Rough is the area beside the fairway where playing a ball is difficult.