Can standard American breeds like Barred Rock and Rhode Island Red be selectively bred to allow them to actually fly? If so, how many generations do you think it would take.
It would depend a little on what you mean by "fly." No chicken is going to fly all day like a migrating duck, or glide in big circles for hours like a hawk. But flying as well as a quail or pheasant or wild turkey might be possible (maybe, maybe not) for some chickens.
For the breeds you named, and any other dual purpose breed, they will never be able to fly very well if you try to maintain the proper body shape for the breed. They are supposed to have heavy bodies (good for meat, but not for flying.)
You may be able to start with them and end up with birds that fly, after quite a few generations. I don't know how many, but probably quite a lot unless you cross in something else. But they will no longer have the right body shapes for those breeds.
If you want chickens that fly, I would suggest you look into small breeds and bantams. Some possibilities: Old English Game Bantams, Sebrights, Hamburgs, Leghorns, Sumatras, Phoenix, etc. You will want light body weight, large wing size, and long wing feathers. Then start with chicks, provide conditions that allow and encourage flying, and breed from the ones that do best.
