HI, welcome to the forum from Louisiana, glad you joined.
Those are not roosters and hens, they are immature cockerels and pullets. When they all mature things will almost certainly change. When will they mature? Who knows? With my pullets, behaviors typically change when they start to lay. I've had a cockerel mature enough when he was about the same age as yours but that is really young. Most of my cockerels have hit that phase by 7 months of age. I've had a couple that have taken longer than that but not many.
The only way you will find out is to let him out and see what happens. Expect him to mate with one or two as soon as he can, that's how a rooster establishes dominance over the flock. What you are looking for is how scared they are of him and whether or not some willingly squat for him. Be a little patient but if you are not happy, lock him up again.
You do not need a new coop, he is not going to be laying eggs. You need some type of shelter to protect him from severe weather. In many climates this time of the year he'd be fine sleeping in trees as far as weather goes so the shelter does not need to be that weatherproof. The biggest concerns are predators and probably the heat. I don't know how big your run is or how predator proof, but perhaps a portion of it sectioned off with chicken wire would be enough. Even in colder climates this time of year an "open air coop" would work. I personally would not worry that much about him roosting. If the choice is between losing his head and sleeping on the ground he's probably better off sleeping on the ground.