After a good nights sleep and looking over my problem I realized that I don't really have to solve the problem of z-fighting along the edges. The reason is this:
Do you see the green edges? well if z-fighting occurs along those edges it's not a big problem because you won't be able to see it since the wrong pixel will connect to all of the other pixels of the triangle it belongs to making it invisible. The points is, we can't see if pixels along the green edges z-fight so we don't have to fix that (lazy me =p).
Now how about the red edges? Well those are the problem, if a wrong pixel pops out from the other side of the cube it's totally out of place on the screen and doesn't connect to any other big body of same colored pixels which makes it really obvious that there is something wrong to the person looking at the applet.
Well we can get rid of the other half of the cube (the one which is not seen) by some type of hidden surface removal. This will speed up things and the red edges won't have anything to fight with therefore solving the problem. I'm not sure yet what the best way is to eliminate not seen triangles but I'll do some research and hopefully come up with an answer.
For those of you interested in solving the problem of z-fighting along the green edges for whatever strange reason ever it probably could be solved by adding a id number to each triangle and biasing the z-fight so that the triangle with the higher id would win even if it's pixel is deeper by 0.0001. I'll implement this when I'm terribly bored and have some spare time to waste. (probably never as it's not needed *.*)