I'm in the camp of wait a year or so before tackling a rooster. If you all were a farm family, used to dealing with livestock, it would be a different situation. But roosters are intact male livestock, just like a buck goat, a bull or a stallion. Just smaller packaging.
Quick chicken reproduction 101
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A rooster will mate with pretty much any hen, of any breed. They don't understand different breeds, as a rule. They know male and female, and the drive to reproduce.
The male mounts the female. Male chickens don't have a penis, no external sexual organs. Each sex has a vent or cloaca, basically the opening under the tail. When the rooster is on top of the hen, she moves her tail out of the way, he curves his body down and they touch vents aka the cloacal kiss. At this point he transfers sperm into her body. He hops off, she shakes herself a bit, that helps store the sperm. One mating can keep a hen fertile a good 2 weeks....but that doesn't stop the roosters from mating hens multiple times a day
. So, when then hen is making the egg, her body uses the stored sperm to fertilize the egg before it's laid.
That freaks some folks out, the thought of eating fertile eggs. Honestly, you'll never, ever be able to tell they're fertile. It takes a very practiced eye to spot the difference, a lot of experienced folks can't even tell when they're actively looking for it. No difference in taste, nutrition, etc.
What about the baby chicks? Eggs have to be incubated at a steady 100-ish degrees for about 3 days for an embryo to even start to develop. So, as long as you collect your eggs and don't allow a hen to set on them, you won't have anything nasty when you crack an egg. I've kept roosters with my layers all my life, and except when I was sick and the kiddos didn't notice a hen setting never ever had an icky egg with a chick starting to develop.
It is a little different than mammals. In my mind, birds are much easier to manage their reproduction than dogs or cats. With mammals, if the male mounts the female, you're pretty much gonna get babies. With chickens, he can mount the hen all he wants. As long as the eggs aren't incubated, no babies.
Like any animals, if the parents are different breed, the offspring are mixed breed animals. It a very acceptable thing in the backyard chicken world, lots of us like to mix and match our breeds
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A hen lays the same color egg all her life. Her egg color is in her DNA. A rooster breeding her doesn't change the color egg she lays. If the rooster has DNA for a different egg color than the hen, their female offspring can lay different color eggs than momma or daddy, it simply depends on the genetics involved.
I know you were asking bout a breed of rooster, and most of us were discouraging about getting one. If you really decide you want to go that route, I'd look at a feather legged breed. Overall, they're more docile than breeds without feathers on the legs. But, that's no guarantee of having a rooster that will do well with littles. There's just something about little kiddos that seem to be a target for a lot of roosters. Children are loud, and fast, and unpredictable and that makes roosters nervous. Nervous animals full of testosterone tend to lash out first, and the kiddos take the brunt of that, usually through no fault of their own.
Since you're just starting on your chicken journey, I wish you and your family the best. They're so much darn fun, you'll be amazed how you lived without them