I recently built a small cow breeder and cooker. Everything works right on it but when I go to cook the cows they light on fire but do not take any damage. On another note, I have yet to have a chicken lay an egg on my residence.
You have to turn animal damage flag on in town. If you type /res info it lists all the flags and /help flags tells you how to turn off and on. Be very cautious of turning on animal damage if you have valuable animals like a cupid pig or promo horses, or any other animal not easily replaced. I had a chicken cooker with animal damage turned on a long time ago on my res, and had a baby horse accidentally kicked to death by another horse.
As for the chickens laying eggs, you have to be within 8 blocks for 5-10 minutes straight. You can extend that 8 blocks to full visible range if you put them in water. When changing the rules for chickens, they just assumed all egg gathering farms would have chickens in water (so not true). To make my auto chicken cooker work, I extended the top section into a large S for water flow and it's been great ever since.
They will only lay eggs when in an active state (in water) or within 10 blocks. This is an EMC only setting. Normal cooldown for ticks applies, chickens will start laying eggs only after you're within range for 5 minutes. The first egg can take up to 10 minutes and then they'll start laying like normal.
Thanks everyone. I had seen that flag but assumed it gave damage ability to a particular person as many of the other flags work in that way. And I will be careful, I have this cooker far away from any valuable animals. Good to know about the chickens too.