I can see your frustration. If you are going to be integrating much in the future I'd suggest building a permanent predator proof shelter. No need to discuss that now unless you have an interest. What you are doing now is too much work.
Do you all think it’s ok to start trying to integrate them into the big girls part? Are they old enough?
We all do these things differently. My brooder is in the coop and my brooder-raised chicks are OK with the adults at 5 weeks. You are not set up like that so you have to do things differently but they are definitely old enough.
I don’t want to put them out when it’s raining since they have no way to get back inside except for me.
Cover the cage with a tarp and leave them out there if rain is your concern. At least during the day.
1. Should I clip the little ones' wings? We don't have a fully enclosed run, we use the Omlet fencing. Don't want them flying over the fence (four feet)
My full sized adults easily fly up to a 5' high roost. I use 4' high electric netting which they could easily fly over if they wanted to. With very few exceptions mine don't. When mine get out is when I have a bunch of cockerels in there. One gets trapped against the fence in one of their fights and can't run away so it goes vertical to escape. It may land on the wrong side of the fence and doesn't know to fly back in. On really rare occasions a hen may do that trying to get away form an amorous rooster.
I learned the hard way to not build sharp corners when I'm setting up that netting. Like yours mine is mobile. A 90 degree corner isn't too bad but anything sharper than that is a good place to trap a chicken. Another mistake I made was to make a narrow corridor leading to a larger area. Like you I had a permanent coop that I had to tie the run back into. Two or three cockerels were getting out every day until I got rid of that narrow strip. So make the area wide.
Your young and old can easily fly over that 4' high fence, the trick is to make it so they don't want to. Since you are integrating it's possible your older hens will chase the young but since they have been side by side for a month and you appear to have enough room that's maybe not highly likely.
and then getting lost! the inside space, the 'nursery' has no access at all to the outside
Since mine are housed there including sleeping in there and they want to be with their buddies they don't wonder off. A bedtime they desperately want to get where they usually sleep. Your nursery is in the coop. They are extremely highly unlikely to wander off and get lost. For the few cockerels that get out I collect them after dark (I know where they are going to sleep, as near their regular sleeping place as they can). Yours may try sleeping in trees, not sure where that would be. Another thought would be to lock all the others in the coop and lure them back in the run with food. Or lock the others in the run and lure or herd it back to the coop.
If I understand the above comments, should I make the first 'contact' encounters outside?
What I would do in your situation after what you have done is to open up that nursery after the three hens are outside and see what happens. That's what I do at five weeks when they have been raised in there. Do it when you can be around to observe and check on them occasionally. You may be surprised how easily that goes.
From outside everyone has access to the main part of the coop, where I have the food and water--and nesting boxes.
I would have separate food and water stations, at least at first. The older sometimes try the bully the younger by hogging the food and water. I'd have at least one station inside the coop and one outside, spread out enough that those hens can't guard them both.
I don't know where those chicks will want to sleep, though probably somewhere in the coop. I really don't care as long as it is somewhere predator safe and not in my nests. I don't try to force mine to sleep on the roosts with the adults, mine don't anyway until they get a lot older.
The first morning or two after I let them all sleep together I'm down there at daylight when they are just waking up to open the pop door so they can get outside for more room if they need it. That's never been a problem but I still do it just in case. I don't see the roosts in your photos so I'm not sure how high they are. With mine, what I normally find is that the adults are on the coop floor while the younger ones are up on the roost where the older can't get to them. They are avoiding the older. During the day they avoid the older too. Mine usually stay two sub-flocks until the younger mature enough to start laying eggs. Then they become one flock.
Good luck and let us know what you do and how it goes. I don't think you are in a bad place.