A broody hen will try to hatch her own eggs, any other hen's eggs, turkey, goose, or duck eggs, door knobs, golf balls, or if any of these are not available she'll just use her imagination. A broody hen should stop laying eggs when she goes broody. She can't afford to use what nutrition and energy she has stored in her body to make eggs since she does not eat much while on the nest. If she is broody, she will not accumulate her own eggs.
You have different ways you can proceed. I let my hens hatch the eggs and raise the chicks with the flock, others isolate the broody from the flock at some or all of these stages. I'll tell you what I do but if someone wants to tell you how they isolate them that's fine. There is no right way or wrong way, just different ways we do this.
Since you have one other hen, start collecting her eggs. Store them pointy side down so the air cell is on the top in a relatively cool spot. I use a spare bedroom at house temperature. I store them for a maximum of a week though others store them longer. We all do things differently. Another option is to find fertile eggs from another source. When you collect all the eggs you want her to hatch, mark them so you can identify them. I use a black Sharpie and draw two lines around them, one the short way and one the long way so I can identify them no matter how they are laying. Put all the eggs under her at one time. As Pyxis said, you do not want a staggered hatch. Those seldom end well.
After the other hen has laid for the day check under the broody hen and remove any eggs that do not belong. These eggs are good to use as long as you collect them once a day. There are two reasons for this. One is to avoid a staggered hatch, but also the hen has to be able to cover all the eggs. If she gets too many to cover, an egg gets pushed out and cools off which can kill the developing chick. Then that dead egg gets pushed back under and another gets pushed out to die. Hens and eggs come in different sizes and different hens have different body shapes. I've had a hen that could only comfortably cover 10 eggs the size she laid, I had a hen hide a nest and hatch 18 chicks.
The 21 day incubation period is a general guideline, not something definite even under a broody hen. I regularly have broody hens hatch one or even two full days early. Others say theirs can be a couple of days late. There are different reasons for this, just don't be surprised if they are early and be patient if the 21 day pass and nothing happens. I
've had broody hens bring the chicks off the nest within 24 hours of the first one hatching, I've had broodies wait for more than 3 days. When a chick internal pips it starts talking to the broody. That way she knows another is on the way so she doesn't abandon the nest too early. Since the chicks absorb the yolk before they hatch they don't have to eat for three full days or more. But once they get hungry or thirsty they start chirping to Mama in a special way, letting her know they need to come off the nest to eat and drink. That's the problem with a staggered hatch, the hen has to decide if she brings the hatched ones off the nest or lets them die while waiting on an unhatched egg.
When I notice a chick has hatched I put food and water on the coop floor where the chicks can get to it. The other chickens will scratch bedding into it, so I put a piece of plywood down on top of the bedding to help keep them from scratching so much trash in it. I put all the chickens on the same feed. The chicks should not eat Layer because of the high calcium content.
I've seen broody hens spoon feed out of adult feeders for their chicks. Within two weeks my chicks are flying up to the adult feeders to eat themselves. I don't have a way to keep the chicks out of the adult food so I go with a Starter of Grower with oyster shell on the side for the hens laying eggs.
Putting food and water out where the chicks can get to them and keeping them clean is all I do. When the hen decides the hatch is over she brings them out of the nest. I've seen a broody hen get chicks out of a 10 feet high hayloft, she said jump and they did. I don't know what your nest looks like but even if it is elevated a bit I wouldn't worry.
My broody hens normally keep the chicks in the coop only for a couple of days, then take them outside where they pretty much spend all day. At night she takes them to a spot on the coop floor to sleep, usually a corner but not always. I don't know what your 6x6 coop looks like, if it is elevated with a ramp there may be some issues initially with the chicks going back in. I consider it a good idea to be down there around dark to see if she needs help with the chicks.
Your coop and run are a nice size for three chickens but you don't have a lot of extra space. I think a broody can raise chicks in it with the other two chickens but you may have issues when she weans them and leaves them all on their own to make their way with the flock. If you decide to separate the broody and her chicks when she is raising them you don't have a lot of room for integration later. You are just as likely to get male chicks as female. You need a plan for those chicks. What are you going to do with them?
To be kind of blunt about it, I think you will be OK when the broody is raising them and after they are adults as long as you only keep a few hens, but you don't have room for multiple roosters. And I think you are in for a lot of drama and even danger to them during their adolescent phase. The coop will probably be OK but I'd give a whole lot of thought to adding some more run space if you are going to keep them during adolescence.
Good luck!
