1) I have a small (smallll) coop that is 3x4 (see photo below... little red guy in background.). If I attached that to the far end of my set up, and put a divider in the run, would that work for a slow introduction? Any tips on how to introduce?
I consider that an excellent plan. House them across wire for a while before you give them an opportunity to mingle. I'm not sure how soon you want to get quail though. I'd house them in that small area until they go to sleep in that small coop at night on their own but for a week minimum, two weeks probably better. You mention a slow introduction, don't get impatient.
When you are ready to let them mingle, try it when you can be around to observe. I open things up in the morning so they have all day to get used to the changes. You might even try the safe haven/panic room concept, have a way the young ones can get back to that small coop area but the bigs cannot follow. That could be cutting holes in your dividing fence or maybe just cracking the gate a little. That way the young ones can get away from the bigs if they need to.
It's possible your young ones will eventually move into that big coop with the big girls at night on their own, I've had that happen. But not often. After they have peacefully fully shared that run space for a month with no safe haven/panic room, I move them in. I close up their small coop so they cannot roost inside and physically toss them in the new coop after it is pretty dark. They are pretty easy to catch and it is too dark inside for the older birds to attack the young ones. I just toss them on the coop floor and let them worry about finding a place to sleep in the dark. They manage, usually forming a group to sleep on the floor. It doesn't hurt them. They are not injured. They are not traumatized. My adults won't let them go to bed on the main roosts with them anyway so they'll need to find a different place to sleep. You can try putting them on the roosts if you want, it wont hurt them either as long as it is too dark for the older birds to attack them. I don't care where they sleep as long as it is predator proof and not in my nests.
Then I'm down there first thing the next morning, as soon as it is light enough for the older ones to be a danger. I keep doing that in the morning until I consider it safe to leave them locked in there together for a while. That's practically always only one or two mornings for me but I don't know what your coop looks like. I typically find my juveniles up on the main roosts in the morning while the adults are on the coop floor. My roosts are high enough that the adults can't peck them from the coop floor. Not sure if yours will be that high.
When I make the move I permanently lock up the juvenile coop so they cannot go to bed in there. Most of the time the juveniles automatically go to bed in the main coop after that first time, but sometimes I have to lock them in the main coop after dark a few times before they get the message. Each brood is different.
Is it true the best time to put new birds (after being gradually introduced) into the coop at night? Do the existing hens just wake up with newbs and think they've always been there?
I guess I do lock them in together at night so they wake up together, but I only do that after they have peacefully roamed together during the day for a month. They are not strangers, they have gotten used to being together. And I do check early the next morning.
2) Are EEs and BOs compatible with grown BR hens? They are just about a year old. Pullets would come at 4 weeks and we would put outside when temps permit.
As compatible as other Barred Rock would be at that age. Don't get hung up on breed. The issues are age difference and whether or not they are strangers.