Cats are temperamental. I once removed a frog that my two cats and the one from next door had been playing with. I'd left them to it but the screams from the poor injured frog led me to take it from them (I had to destroy it, awful). My two cats went up on the garage roof and refused to come in or days. They wouldn't eat the food I put out for them but when I bought an expensive cat food, down they came. They were disdainful.
As for introducing dogs, I agree with the advice to avoid a terrier type and definitely not a sight hound, though both breeds make brilliant pets, their chase and kill instinct is strong.
My experience of cats and dogs living together is that the cats rule the roost. A friend who fosters dogs has put a child gate on the stairs, the cats are allowed upstairs, the dogs aren't. It works well.
Some dogs and cats form friendships and snuggle up together, hard though that is to believe. My friends 3 cats take over her 2 dogs beds, the dogs back off.