If for meat birds you intend to raise Rocks or other dual purpose birds to broiler weight, no reason they couldn't share a coop. If, as I suspect, you are talking about Cornish Crosses for meaties, they are completely different creatures and should share a coop with your layers only if you can partition them off to keep them separate.
As was said above, one concern is the meatbirds taking over to the detriment of the layers. For comparison, this years brood hatched the same day, eight weeks ago last Friday. The layers are thriving but still small enough to cup in one hand easily and perch on two of my fingers. By comparison, the meat birds lost their feathers on Monday, and dressed an average of 5-1/4 pounds apiece, with a few of the larger ones near six pounds. The smaller birds wouldn't stand a chance competing against them if they all had access to all the same feeders.
And, still on the matter of feed, they'd need to be separate because they need to be fed differently. It goes without saying that all birds should have free access to water at all times, and the same generally holds true of feed, EXCEPT for birds like the CornishXs who, to limit problems with their legs and cardio-pulmonary weaknesses that result from their rapid growth, are generally fed 12 hours on/12 off.
As for the turkeys, I don't know how to advise. I'm only vaguely aware of the issue of blackhead developing in turkeys that are around chickens, but there are some posts around from folks who have successfully run them together. Maybe try the search engine on the site, searching on "blackhead", for some guidance on that issue.