I think it depends on the animals. I had my cat for 9 years and then got the 3 dogs. My dogs are 3 different breeds and we got 2 of them as adults one was 2 the other 4yrs they all do well with my cat. My cat took his time accepting the does but when we tried to add another cat he refused to accept it.
First of all, does your father want a dog? Is he steady on his feet and watch where he is walking? Dogs can be great companions but can be a hazard fir an older person, ie, tripping, jumping, need for care, etc.
I have had cats and dogs raised together but most from a young age. I had Shekties when I had cats. A baby gate was usually kept up and the cats could just jump over it.
I also helped with Greyhound rescue for some years. We adopted one and I had cats. I did have to be vigilant at first because he would go after the cats when they ran, eventually, that stopped. Most rescue groups cat test the dogs. They can be nice companion dogs and most love to take walks.