An 8 x 14 run isn't huge for integration but along with the coop if could easily work. The coop and run together is your space system.
Either way should work. Your young ones may instinctively be afraid of the older ones or, if they invade their personal space, may get pecked and quickly learn to avoid the older ones. Don't be surprised if the young ones are in the coop when the older are outside. It can work the other way but usually the older like to be outside. Sometimes they intermingle a lot more than I make it sound, but usually mine avoid the older ones. Having clutter in the run can really help. So can having separate feed and water stations. In your case I'd have food and water in the run and in the coop.
My goal in integration is that no one gets hurt. It's that simple. When the young ones mature into adults they should all become one big happy flock. That will happen, but be patient and let them get there at their pace.
I'm not sure how big your coop is or how it is set up. With only 6 total chickens it may not be that big. At night my juveniles tend to not sleep that close to the adults. Like everything else there can be exceptions to that but mine are pretty consistent about not sleeping that close to the adults. In a small coop that may mean the juveniles don't want to be in there at all. When I'm integrating I don't care where the juveniles sleep as long as it is not in my nests and is predator safe. I try to not forcibly lock mine in the main coop with the adults until they prove to me they can coexist during the day without drama. When I do I make sure I'm down there when they are waking up to make sure it is going OK.
My brooder is in the coop, so mine grow up there. When I'm ready to let them out I let the older ones out first, then open the door to the brooder. That's early in the day. While I keep an eye on things, I pretty much leave the rest up to them.