It's normal. Chicks aren't members of the flock yet, so if a mother hen is not able to protect them from her flockmates, or if you just mix chicks in with older birds, they will sometimes kill, or at least terrorize the younger chicks. Always raise chicks separately if they don't have a mother hen. If they do have a mother hen, watch and see how well she is able to protect them from flock mates, before leaving them and her with the flock.
I'm not too surprised about the cat, either, if it was very small, and especially if it was a kitten. If cats behave aggressively towards them, they are usually somewhat afraid of it, but if any small animal behaves submissively, they will sometimes attack. They might start picking at and eating the dead cat, too, as chickens are omnivores.
As for no eggs from 8 month old chickens, try adding store bought eggs to the nestboxes and see if they disappear. If the chickens are not eating them, some other animal might be stealing them.