After a week or so of nervousness, your chickens will get used to their new surroundings, and their new neighbors - human and otherwise! If you introduce your dogs to the newcomers, they will soon stop wanting to chase the birds on sight. Or, at least, they will only want to play, rather than chase and kill.
If your dog is well trained and listens to you, you should be able to make the introductions without too much panic and excitement. Use a lead for the first introductions. Don't underestimate the courage of a tame chicken - they will soon be scratching under the dog's feet without fear, and will let him know, with a sharp peck, if he's getting in their way!
Your chickens should be able to share garden space with your other pets.
Cats, too, will soon get used to the novelty of chickens in the garden. Unless you have a very small breed of hen, the birds will be too big to register on the cat's radar as "food" (apart from chicks, which should always be protected from cats).
Chickens and Ducks
Chickens and ducks mix well - as long as the ducks are all female!
Drakes - male ducks - will try to mate with anything that has feathers. Their violent attempts will cause distress to your hens, who will usually crouch down once caught, instinctively flattening themselves in the 'mating position'. As the drake tends to grip his mate with his beak, a hen's comb can easily be torn, and she will lose neck feathers too.
So, no male ducks! Female ducks, however, will waddle around in their own flock and pay little attention to the hens. They may compete for scraps, but usually with less violence than a group of hens fighting for the same treats!