Cats by nature are independent & solitary creatures and you must be cautious whenever you introduce your cats to other pets. You must make sure that there is no hostility or unease among them. If you feel that the 2 animals are not getting along too well under your observation, there is no way they will grow to love each other in your absence. This is the only way you can find out whether your cats can live with other pets.
This especially applies to cats and dogs. Ideally, these two species should be introduced to each other when they are kittens and puppies. Then, they learn to live with each other, and probably even grow fond of each other. However, for older animals, this relationship comes with more difficulty, and you have to chaperone the first few meetings and protect the more vulnerable pet.
Dogs by nature are predatory creatures, and will not tolerate the new pet if they are aggressive towards the species. Likewise, if the residential cat is a bully by nature, it will attack the puppy whenever it gets an opportunity. In these situations, the puppy/dog may learn to steer clear of the dangerous cat's way, or the kitten/cat may live in caution and anxiety of the dog; however, both these cases are not desirable to you as the owner.
One solution is to find out the nature of the two species that you want to keep as pets. Is the cat sociable and the dog meek? This relationship will probably work. Is the dog aggressive and the cat hostile? This may probably not work at all. Also you could introduce the two pets on a trial basis; keep them under observation for a few days, and if you think they have started warming up to each other, keep it going; else give it up before someone gets hurt.
It becomes even more difficult when it comes to the question of keeping cats and mice together. It is in the nature of cats to hunt mice; there is nothing you can do to change this fact. Therefore, if you want both these species as pets, you need to take certain essential precautions. For instance, the mice should consistently be caged when the cat is anywhere approximately, and the cage should be sturdy and kept at a height.
The same policy should be adopted with fish and birds. The lid of the fish tank should be strong and kept tightly closed to avoid a cat from making a meal of the fish. Similarly, the bird feeder or a bird cage should be away from a cat's reach. Anything else is inviting disaster.