We keep them in so many different conditions that there can’t be one magic number that is required for all of us. Some chickens take confinement better than others. We manage them differently. We have different weather or maybe work or school schedules that put restrictions on us. And it’s not about coop space by itself but how much total space is available to them and when it is available. It doesn’t matter if it is the coop by itself, the combined coop and run, or maybe coop and free range as long as when it is available is considered.
A coop that size should work for you with those birds provided you use it only for a safe place to lock them up at night and you commit to letting them out about as soon as they wake up every day of the year, even if you want to sleep in on a weekend, when you have the flu, or when you are on vacation. That’s if the weather allows them to get out. If you live where weather keeps them in the coop for long periods of time, you could have problems. Don’t just think of those perfect days but consider when you have really bad weather. Some people do keep that number of chickens in that space but it takes a certain commitment every day if you crowd them like that.
You can probably come up with a way to get a nest or two in there but you might have trouble fitting in feeding and watering areas that they don’t poop in from the roosts. Layout can be kind of rough in those small coops. Some people manage though. If they can be out each day, you may have to feed and water in the run.
One of the risks of overcrowding birds is that they can become very aggressive toward each other. This can lead to feather-picking, fighting, all the way to cannibalism. Commercial operations have proven that you can keep chickens in very little space but they often have to take extraordinary measures to keep them from harming each other, like beak trimming. Most of us want to do better than that.
The way I envision your set-up, if you are willing to make that commitment to let them out every day of the year, it would probably work out for you. The coop and run together gives them plenty of space. That little bit of free ranging is even better.
One of the handicaps is that you don’t have a lot of flexibility in how you handle problems. Say a predator finds its way into your run. You can’t leave them locked in the coop while you go to work but you have to deal with that problem immediately. Or maybe you have a weather-related issue?
Another aspect of flexibility is that if you want a hen to raise chicks with the flock or you want to integrate more chickens later, that goes a whole lot easier if you have some extra room to work with. I feel that most of the broody hen problems or integration problems you read about on this forum are due in part to a lack of extra space.
I find that the more I crowd them the harder I have to work. Not just responding to things but think poop management. The more crowded they are the more concentrated the poop is. You can handle that; you just might have to work harder at it or come up with a strategy like a droppings board.
I always suggest providing as much room as you reasonably can. I find I have fewer behavioral problems, more flexibility in dealing with things that happen, and I have to work less hard. It’s not that you can’t get by with less space, just that it is less enjoyable.
Hope you get something out of this long rambling that helps. Good luck!