I am not quite sure where your confusion lies, but I will try.
Chicken fanciers breed for "breeds", not for "colors". Chickens are bred to try to get the bird closest to the Standard of Perfection for that particular breed.
The Standard of Perfection calls for specific colors, and so, in order for the bird to be as close to the Standard as possible, the chicken hobbyist pays close attention to color.
If you cross chickens of 2 different breeds, you are breeding mutts and there will be no standard for them, so you can do anything you want and call it good.
If all you want is to breed for colors, there is nothing wrong with that. Lots of people like their mixed breed flocks.
Temperament is inherited, so you decide what temperament you want, and breed the chickens with that temperament and remove the ones whose temperaments you don't like and don't breed those.
If you want really large chickens, you choose the biggest birds in your flock and breed those. You remove the itty bitty ones and don't breed those.
If you want tiny chickens, you use the little ones for your breeding stock.
If you want lots of eggs, you choose the hens that lay lots and lots of eggs and you use them for your breeding stock.
Chicken fanciers breed for "breeds", not for "colors". Chickens are bred to try to get the bird closest to the Standard of Perfection for that particular breed.
The Standard of Perfection calls for specific colors, and so, in order for the bird to be as close to the Standard as possible, the chicken hobbyist pays close attention to color.
If you cross chickens of 2 different breeds, you are breeding mutts and there will be no standard for them, so you can do anything you want and call it good.
If all you want is to breed for colors, there is nothing wrong with that. Lots of people like their mixed breed flocks.
Temperament is inherited, so you decide what temperament you want, and breed the chickens with that temperament and remove the ones whose temperaments you don't like and don't breed those.
If you want really large chickens, you choose the biggest birds in your flock and breed those. You remove the itty bitty ones and don't breed those.
If you want tiny chickens, you use the little ones for your breeding stock.
If you want lots of eggs, you choose the hens that lay lots and lots of eggs and you use them for your breeding stock.