I was trying to find to post where you talked about how you introduced your dog to cats.. Unfortunately I just can't find it scanning through.. So I apologize that you've already talked about it, but please, how did you introduce them?
First off, every dog that I ever had except one was raised with cats. When they were little puppies, they would try to chase the cat and the cat swatted them on the nose. They developed a deep respect for cats and they never noticed that they weigh 100 pounds and the cat only 6 pounds. This is by far the best way.
I did have a dog that was given to me as an adult. He killed and mutilated my daughter's beloved cat. My first reaction was to put him down, but I decided to try to train him. I would take him out on the leash to start with. If he even walked toward the cats, I would jerk his dang head off. If he chased them, I took a riding crop to him. Since he was so stubborn, it got so that basically even looking toward the cats earned him a whipping. He did leave our cats alone. He didn't like it, but he learned to out of respect for me (or the riding crop). I never totally trusted him alone in the house with the cats. He was a fairly aggressive dog anyway, so he was in the pen unless I was with him. He was a great horseback riding dog. He ran in front of the horses and kept the path clear of wildlife.
I will never accept a grown dog that is not totally cool with cats, horses, chickens, and children again. It was too much trouble and too much heartbreak. If I were going to try it again, I would use an electric collar. But I'm not going to. There are too many nice dogs available.
Puppies learn so much easier.