We live in a multispecies house. There are always dogs, cats, parrots and rabbits around and the range of animals that has come through the door is staggering.. most birds (chickens, finches, budgies, doves, quail, large cockatoos etc) most small "pocket pets" (hamsters, gerbils, rats, mice), turtles, lizards, chinchillas, guinea pigs and probably more I'm missing ...
It is not always peaceful in our wild kingdom but generally it's much more laid back then someone who was in a single or dual species home would suspect.
Introducing cats to dogs and dogs to cats is a fairly straightforward process.
Puppies or kittens make the job much easier as they have no preconceived notions that anything might be exciting about this. If you aren't sure of the history of either in the pairing go SLOWLY.
New dog in house ....
unless you can directly supervise them both keep them separate. Period. Possibly forever depending on the size of the dog and how things go. Here Sampson (the resource guarder) will likely never be out alone with the cats. Accidents can and do happen in a heartbeat. You being there may not be enough but odds are good you will see tension rise and be able to deal with it. Sally will occasionally bounce at a cat but she never threatens them or makes contact. The cats here are very dog savvy and rub up on all of the crew here. We make the choice to have the small dogs and Sally loose in the house with the cats when we are out that could change at any time. Our house is set up so there is a dog free zone - I actually think that is a really big part of why the foster cats adapt so relatively quickly to the chaos!
When you first introduce cat and dog it is much easier to have two people, a relatively small space, something high for the cat to sit on and some time and patience. The two people means that each person is assigned to keep one animal calm and happy. Cat is up on high spot, dog is brought in. If there is no reaction that's perfect. Both are praised and rewarded. Dog sniffs cat gently. Cat bristles but stays calm? Also perfect. Praise, reward. Dog tries to leap at cat? Not terrible but back dog up til things are calm. Then reward. Don't bribe the dog or cat. Don't shove them together for ages if either are stressed. Many short positive meetings are the goal. There is no point in jerking, smacking or punishing anybody if the meeting isn't calm. A deflection, then a redirection and then a retry with a different plan in place is the way to go. The last thing you want is either the dog or cat believing that the other species causes stress or bad things to happen. The best things should happen when everybody is calm together!
If you do nothing else to get ready for the introduction read the story of Song and the Sheep. Patience is a key to long happy relationship building and ideally that's what you want, right?
Questions? Please ask - I have done this HUNDREDS of times - with many dogs and many cats, many puppies and many kittens including cat killing dogs and dog attacking cats. You can negotiate a peace. It may not look like the pictures on this post
but it can be done.