Yes, you can keep chickens, ducks and geese together. All of my animals are together. As far as can they be brooded together........... If I am understanding your question, you're asking if once they become broody, can they all be together. Well, I don't know about that. I think that would be difficult to do. When all of my animals go broody, they are setting in different places. I have outside brooders for the chickens. The ducks are have separate brooder quarters (dog kennels, etc.) and the geese also have different brooder quarters (dog houses). My ducks and geese still need an area large enough to bath in and place their heads/bill inside and unders. The dog kennels are large enough where I can put large containers of water inside. The ducks and geese get a little more agitated as far as visitors going into their living quarters and putting food and water in. The chickens are really not bothered. They just go with the flow. I'm not sure if that helps you. This is just my experience.
I think the OP is asking if you can raise chicks, ducks and geese together in the same brooder. I raised chicks last fall and am now raising a few ducks. The dreucks are very messy with their water, and I would assume the geese would be the same way. Chicks need to stay dry. So unless you had a large brooder that allowed for a wet and dry area, I wouldn't advise of brooding them together.
I have all 3, but brood, house and raise them separately. Ducks and geese have such different needs from ground birds that it seems a little nuts to raise them together. Waterfowl love to play in water, so they are constantly mucking up the waterers and leaving the run a sodden mess. Chickens hate water as a rule, so they have serious issues with this behavior. Chickens also roost high up, so the ducks (that don't roost) usually end up parked underneath them and get covered with poop all night. The only need they have that is similar is feed, so there are generally no problems there, but that is where the similarities end.
I think my chickens would divorce me if I tried to house my waterfowl with them.
I started out with my ducklings and chickens in separate brooders. The ducklings like to play in the water so much that they make the brooder a mess. Chicks keep it relatively clean. When they got bigger I moved them into the coop together. Again ducks are messy so the coop had to be cleaned often (like weekly). I finally decided the ducks needed their own place. This was partly due to the mess but more so because we let the ducks out to spend the day on the lake and it was a fight to keep the chickens in the run and letting the ducks out. If you don't mind extra clean up, it's fine. If fact when my goose went broody and was determined to hatch a duck egg, I gave her 1 egg and moved her into the chicken coop to keep her nice and safe. She did snip at the chickens if they got too close but caused no problems and she successfully hatched her "guckling"
Ducks are pigs. Water all over. I have two ducks with my chickens and cant wait to move them. I am short pens for now so waiting to get ome moved and then move them by themselves. Muscovies aren't to bad with chickens as they dont require a big pond. I just give them a kiddie pool to play with.