So many good questions but unfortunately they require opinions. With chicken behaviors there are a lot of if's and maybe's. Each is an individual with its own personality. The only thing consistent about chicken behaviors is that they are inconsistent. I do believe that how much room they have has a big effect on behaviors, but you can always get exceptions.
Do chickens know their own? From my observations, usually not. Part of that may be how I manage them, giving them a lot of space and freedom to manage their own affairs. Once the hen weans her chicks they are generally treated like any other flock member, though it can take a little time for this to fully kick in. Several people on this forum have posted exceptions to this.
How big is the new coop? It sounds like some of your issues in the old coop may have had to do with space. Also, how big is the run and can you lock them all in the run to keep them out of trees? I assume the new coop has access to the run? Does the old coop have access to that run? The more I know about your situation and what you have to work with the better I can comment specifically. A photo or two might help.
Another issue is that the hen sounds feral. She may be hard to break from sleeping in trees and she is going to decide where the chicks sleep. How badly do you want to keep that hen and try to tame her? You may be better off taking the chicks from her and raising them yourself. The solution needs to accommodate your wishes and desires. You can probably manage to tame her but it might be easier to let her return to the rooster and flock. If she is hatching chicks there is a rooster.
Assuming the coop and run are big enough and the run will keep them out of trees, you have options. It sounds like they have been roaming together for a while. You can keep them confined to the new coop/run. At night move them into the new coop after dark and lock them in there until early the next morning. Be down there early to let them out until you are comfortable they are not fighting in the coop. This is fairly close to how I manage my brooder raised chicks.
I let my broodies raise their chicks with the flock so they can handle integration. Often they sleep in the main coop from hatch. But if the main coop is getting crowded I have the broody and her chicks sleep in a shelter in the run until she weans them and goes back to the main coop on her own. On rare occasions she will move the chicks back to the coop before she weans them but usually she leaves them on their own and they return to that shelter at night. They have been roaming with the flock during the day under the broody hen's protection basically since hatch. There are factors, but I typically move them to the main coop (if I have room) at 12 weeks.
Neither of these totally fit your situation because mine are not feral. Good luck!