How cats and dogs get along.

Here's Cooper, Elvis and Daphne.
I have read - and observed - that one component in successful dog-cat harmony is the dog regarding the cat as a Higher Being. In other words, somewhat higher in the pack order than the dogs. Pack order is fluid, of course and not as cut and dried as some make it out to be. A dog may relinquish food or a bone to another household dog, yet not allow head-over-the-withers or dominant humping. Dogs seem to work out these rules and come to respect each other, for the most part. Dogs being social animals learn to get along just fine with other species.
I have watched my dogs, over the years, stand back and wait respectfully if the resident cat is eating something at floor level. Even though a 100lb dog can quite easily push or scare the cat away from the food. Smart, dog-savvy cats rule, and they know it. An unconfident cat won't command the same respect from most dogs, and a cat who runs away shrieking will usually trigger a prey response.
This isn't to say all dogs get along with cats. Terriers and sighthounds are often cat-aggressive by immutable nature. Most of the dogs I've owned will regard a strange cat as an interloper that needs to be killed - or at least chased away. They have to be taught that the cat is part of the house pack. For this to work, they also need to have absolute respect for the human in charge. The cat is my property, not theirs. Just as books, furniture and shoes aren't supposed to be chewed up, nor is the cat!
A wise cat also learns limits from the dogs. Elvis has learned not to go near either dogs' head when they're eating or enjoying a particularly nasty chew bone, or he gets a warning growl and lunge. These are not intended to harm although it looks scary; just to warn. An honest dog will warn first and escalate threats as needed. Most Rottweilers are very honest dogs.