The cork points in the direction of acceleration because the water has a bigger density and moves back instead of the cork. If there was no water in the bottle, the cork would do exactly what you do. When ever you look at a free body diagram, the dot moves in which ever way the acceleration is and thats the same with the cork in the bottle.
1. If you walk in a direction and speed up, you make it go forward in the bottle, but when you suddenly stop it flings backward to where you are. To make it come to you, you would have to do the opposite what you did the first time. Since which ever way you are walking it seems to go away from you, but when you walk backwards it moves with acceleration and the points to where you are. The cork is just imitating the free body diagrams we were using all week, by moving in the direction of acceleration. There is no where else that the cork can go other than the way of acceleration because the water goes that way instead.
4. The cork pointed to my hand when I had swung it around at arms lenght like a fool. It pointed to my hand because I was making it accelerate. When ever you stop though, its always going to go in the opposite direction of acceleration and then back to the middle. Since you stopped moving it, the water goes back to its original place and the cork then rushes to the other side and then back to the middle.