From what I've read about Kosher Kings I'd treat them like Rangers. I think that's what they basically are. I don't know what has gone into the KK breeding other than they are based on dual purpose breeds. Sometimes breeders put things in the parent flocks that don't affect the actual meat birds but will show up in the offspring. So you can get some surprises when you hatch eggs. I'm specifically thinking of dwarfism, I don't know if the KK's have that or not. If you do get strange results like that come back on and we can discuss it. I would not worry about it ahead of time, worry is interest paid before it is due, but if something like that happens it can be dealt with. So if you see something like that don't panic.
Many people on this forum have kept Rangers of various types for their breeding program. Some breed Rangers to Rangers, some breed Rangers to dual purpose breeds. With all the "interesting" people on this forum some have probably tried breeding Rangers to Cornish X. I have not bred Rangers myself but the point is that there is a lot of experience on here with Rangers and on some threads. You might look for some of them and see what they say about restrictive feeding. My concern is that if you let them eat all they want they either get too big to breed or they die by eating themselves to death.
But for your basic questions, yes they are very capable of breeding if managed properly. Will the babies produce the same results? From what I read they are hybrids so you will not get consistent results. Some should be as good as their parents, some will not. But they all should have the genetics that creates fast growing chickens that taste good. The main difference is that some will be bigger than others or may have slightly different conformations.