new member needs to know

Hi, I just joined Back Yard Chickens today. I am going through my first winter with my first flock. I bought several new chicks this spring. We have a nice hand made chicken coop and we have just recently began to leave the fence open so our chickens can go all over the yard looking for food. We used to keep them pinned up because of our Pomeranian. He bit three chickens and broke the necks of both guineas. I decided to give him away. This spring I plan on buying more guineas and a couple of peacocks. I tried to let one of my hens raise a few baby chicks. She sat on those eggs for a month or more but other hens would lay beside her and she couldn't keep all of the eggs warm. One day my first egg cracked open and I found a dead baby chick sitting in the shell with the top off. I think it froze. I took the rest of the eggs and cracked them open to find five baby chickens and the rest of the eggs were yolks. I think next time I'll mark the original eggs with a marker and remove all the additional eggs laid. If I don't get a live chick that way then I'll use an incubator. My chickens love to eat my cat's food just as much as their own food. Will this spoil the eggs?