I am attempting to create a simple game in Java using Processing which utilizes paths and rotation in order to create an element of the game.  In this game, lines will appear at random, on click, at different positions along a circular path. You, the player, will control a circle with your mouse positioning and click in hopes of avoiding the randomly generated lines.

The only problem is that I do not know how to attach shapes to a path (specifically an elliptical path) in random order.  All of the shapes (lines, in this case) will extend through the screen.

Which functions should/can I use to create this effect?

What specifically is it you have problems with? Finding the position where to create your objects?

To position something on a circular path, you can use the sin and cos function. Just assign a random number between 0 and TWO_PI (doesn't matter if it is out of this range either, but just to keep it simple) and use that value to define the point on the path.

Example: Code:

void setup(){
  size(500,500,P3D);
  frameRate(2);
}
void draw(){
translate(width/2, height/2); // place the circle in the middle of the canvas
float randomNum = random(TWO_PI);
float xpos = cos(randomNum)*200; // 200 is the radius of the circle
float ypos = sin(randomNum)*200;
translate(xpos,ypos);  // move to the point on the circle
box(10); // draw something at that point ...
} 

The code is just an example, but I think it should work, although untested. Is this what you were looking for?
Edited: