I do not know what you mean by the cock bird with "the worst disposition". If you have to watch your back with him, he is not worth messing with.
Improving egg size is best done by egg culling. Once you feel established, set a standard. Do not set any under a certain size. Gradually raising your standard from year to year as you see improvement.
Continue to develop your eye for good type, and learn all that you can about the color. You cannot intelligently do it without a Standard and a scale.
You will notice that your flock has certain tendencies, and see how it is inherited. Some faults you will not be able to tolerate, and some you might get away with using once or twice.
If you can find someone to give you some insight as you move along, it would be helpful. I did not have that, and learned a lot the hard way. Which is expensive.
In a couple generations, you will know what you have, what you need, and what you need to do. Don't get too complicated until you get a feel for them and the process. I did not do that, and complicated it more than it needed to be. I still have that tendency. Enjoy the birds first. It all comes with time.