We built it in a place that doesn't get much sun so it's really muddy.
Is the problem sunlight or that the ground just doesn't drain? The most critical issue to me when building a new coop/run is that it be in an area that drains well or fix the area so it does drain. I'll give you a link to an article by a lady that said she lived in a swamp, so has some experience dealing with muddy runs.
Pat’s Big Ol' Mud Page (fixing muddy runs):
https://www.backyardchickens.com/articles/how-to-fix-a-muddy-run-chicken-coop.47807/
1. when can i put the new birds outdoors
You don't say where you are or what your weather is like, but it's hard to imagine a situation north of the equator where they could not have been outside three weeks ago if weather is the criteria. Most chicks are fully feathered at 4 weeks, 5 weeks at the most, and can handle really cold weather. It's also pretty challenging to imagine a situation south of the equator this time of the year where 8 weeks would be a problem.
2. How do I introduce the two sets of birds to each other and not cause a fight!
You are dealing with living animals so no one can give you any guarantees about their behaviors or how they will interact. Eventually they will sort out the pecking order (I assume your three are pullets?) but until pullets start to lay mine pretty much form a sub-flock and avoid the adults as much as they can. That's during the day and at night on the roosts. During the day they will sometimes mingle quite a bit but that varies from one group to the next. Each flock has its own dynamics and, since they are living animals, can behave differently. I don''t know how big your back yard is or how it is laid out, but I'd imagine it is plenty big for 10 birds.
My brooder is in the coop so my situation is somewhat different from yours. Since my chicks are raised with the flock, I have weather the adults can spend practically all day every day outside, and I have lots of room inside and out integration is easy. You are in a different situation but it doesn't have to be hard.
What I'd do is house your chicks separately from the adults but with access to the backyard and where they can see each other. That could be a separate coops or partition a space in the coop for the chicks if you have room. House the chicks here until you are comfortable they will return at night. Try a week. If that doesn't work try housing them longer or moving them into their sleeping space physically after it gets dark. Eventually they will get the message. The main thing is that the chicks do not try to sleep in the same space as the hens at night. As they are settling in for the night is when mine are most brutal to each other and, depending on your coop, having them locked in a tight space when they wake up in the morning is also a risk.
I can see two ways to test the chicks. Let them alternate with the adults as to who gets to range that day or lure the hens into their coop at night a couple of hours early and let the chicks out then.
After you are comfortable the chicks will return to their own space at night, let them mingle during the day when you can observe. It's always possible one of your hens is a true brute and will seek out the chicks to do real harm, but with a lot of space that is pretty unlikely. Still, observe. When they first mingle the chicks might approach the hens. If the chicks invade her personal space a hen will likely peck a chick. The chicks should run away and very quickly learn to avoid the adults. As long as an adult doesn't chase them down or trap one against a fence or other tight space, life should be good.
After the chicks have roamed with the adults about a month, try moving them into the same coop at night. After dark and the adults are on the roosts, lock the chicks in the main coop and lock their old coop so they can never return. They are unlikely to sleep on the main roosts with the adults unless you have a lot of roost space well spread out, they chicks are still in avoid mode. As long as it is not my nests I don't worry about where mine sleep when I'm in this mode of integration. When the three start to lay they will move to the main roosts with the adults. If they do start to sleep in the nests get back to us, there are different ways to handle that. When I do this I'm down there at first light until I'm sure there are no issues with them locked in the same coop with the adults. It never has been but it's still good practice to check a time or two.
The hard part might be giving the chicks access to the backyard while keeping the adults confined. A lot of this depends on how big your facilities are and how they are laid out. We each have unique situations with unique flock and layouts, no wonder we get different results.
Is this overkill? Absolutely. You wanted a safe method and I can't design a safer one. Many people don't do nearly this much and don't have problems. I try to follow the general principles of take my time, don't get inpatient. And a big one, do not try to force them to be together in a tight space until they really get to know each other well. Even then, let them do as much as they will on their own and on their schedule. If they don't want to roost together, don't force them. If the young want to avoid the older, let them. When they mature you will have one happy flock. Until then it may look like two separate flocks which is fine. My definition of a successful integration is that no one gets hurt.
Good luck!
