For all of this it would really really help to know what kind of area you are working with. What is the size, in feet, of your various coops and runs? How are they connected? What does the inside of your main coop look like? Photos could be a big help.
I introduce my chicks straight out of the incubator or from the post office but that's because my brooder is in the coop. They grow up with the flock, just across wire. At five weeks I typically open the brooder door and walk away. That's it, integration is done.
Will this work for you? I seriously doubt it. My coop is relatively big for the number of chickens I have (8' x 12'), I have a couple of extra shelters I can use if I need them, my main run is 12' x 32', I have an area about 45' x 65' in electric netting, and my weather allows the adults to be outside practically all day every day except when they are sleeping or laying eggs. My coop is sort of cluttered with hiding places and extra roosts for juveniles. I don't care where mine sleep as long as it is not in my nests and is somewhere predator safe. I don't try to force them together so they have to learn to get along. I give them as much room and freedom as I can so they can manage that themselves. With your number of chickens I get the feeling that your space may be limited. Hopefully I'm wrong because tight space can make integration much harder.
Knowing what you have to work with can help come up with suggestions for your situation. We can give you generic stuff but without knowing what you have to work with that may actually hurt.
When I have different age groups the younger tend to form a sub-flock and avoid the older ones. The mature ones often peck the immature ones when they invade their personal space so they quickly learn to not invade their personal space. I have enough room they can do that. If your room is so tight they can't avoid integration becomes much harder.
As for them sleeping in the coop, it would be good to know what the coop looks like and what the roosts inside look like. There may be something obvious about the roosts and coop. Even how much light you have inside can make a difference. If your window is such that it's really dark they may not be able to see well enough to go to bed. Or maybe you need to remove the roosts in the run.
Typically though that is a matter of habit. They are used to sleeping outside and you need to break that habit. The way I do that is every day after it is dark enough for them to go to bed I lock them in the coop overnight. Be consistent, do this every night until they start going to bed in there on their own. This may take a few days, it may take a few weeks. Many people like to set them on the roosts when they do this, I just put them on the coop floor and let them work it out.
Typically my immature chicks will not sleep on the roosts with the adults until they mature. Where they roost is a pecking order thing. The ones higher in the pecking order sleep where they want and choose who they will allow to sleep close to them. They can be (and often are) brutal to immature chicks that try to sleep with them, so mine sleep other places. If your older ones are sleeping in the run that may be in the coop. If your older ones are sleeping in the coop the juveniles may sleep in the run. If your coop is big enough and they have a safe place to sleep, both groups may sleep in the coop.
I agree with Aart. Sometimes you can mix chicks a week or two or even more apart and it is not a big deal, they get along great. Sometimes it is a disaster. For different reasons it's usually easier to introduce more the same age than just one at a time. Since you are only adding four total I'd be kind of nervous.
Many of us do what you are talking about regularly. For some of us it is easier than others, but you can do it. We just need to know what you have to work with to give you suggestions that might work for you.
Good luck!
