I read that as 4 feet square so 16 square feet. With a 5x3 (15 square fert) lower section. That's still tight for integration, let alone for 13 chickens total. I don't think it will be big enough.
I let mine roam in the run (I have over 2,000 square feet on the run, you have 5,000
) until I'm convinced they can peacefully coexist without the hens chasing them and beating them up. Also the chicks need to learn to avoid the adults, don't go invading their personal space. I usually let mine roam the run together for four weeks before I start putting them in the main coop to sleep. Nothing magical about four weeks, I'm just not in a hurry to put them in. I could do it earlier.
My main coop is 8 feet x 12 feet. The adults sleep on the roosts, the chicks find other places to sleep. They do not share roosts until the younger one start laying. As long as they do not sleep in my nests and where they sleep is predator safe I don't much care where they sleep.
I'd keep doing what you are doing as far as them seeing each other. Don't try to shoehorn them into a tight space and things should go well. But I'm not comfortable thinking about you trying to get them to all sleep in that two level coop.
I let mine roam in the run (I have over 2,000 square feet on the run, you have 5,000

My main coop is 8 feet x 12 feet. The adults sleep on the roosts, the chicks find other places to sleep. They do not share roosts until the younger one start laying. As long as they do not sleep in my nests and where they sleep is predator safe I don't much care where they sleep.
I'd keep doing what you are doing as far as them seeing each other. Don't try to shoehorn them into a tight space and things should go well. But I'm not comfortable thinking about you trying to get them to all sleep in that two level coop.