Some breeds have a high prey drive; and yet an individual dog in that breed "may' be OK with cats.

Some breeds like bichons have little or no prey drive. And yet, my first fosters thought cats were edible! LOL we relocated them to another foster home within 48 hours, ha haa.

So you should consider the breed type, IMO.

A shelter, a rescue: they can temperament test a dog for you, let you know if it is OK with cats. This may be more difficult to do with a puppy, as they are still developing; so you go more by the breed type. Yes, a puppy growing up with a cat is going to adjust to that cat very easily.

And by the way, the CAT will be the boss! Here, even the smallest cat at 6 pounds can easily boss the big 22 pound bichon around, lol. The cat will swat the nose once, nails retracted. With my crew, depending on the cat, the 2d time, the nails may or may not be retracted. Any dog fool hardy enough to get too close when the cat has made it clear she doesn't want to play gets the nails out on the 3rd swat and believe me, that is the END of the dog trying to be boss, ha haa.

Don't get me wrong. I have pics of my cats and dogs snuggled together sleeping. They do get along here. But as to who is in charge? I have a video of my Marlin whining, he wants to drink at the water bowl. My RB Vita was at the bowl, and when she finished, she sat and . . . stared at the ceiling, roomed every hair in to place, just a real tease. Marlin would NOT approach that water bowl until I moved her, ha haaa.

A few thins, which you mom may or may not know:
1. The dog should never be allowed to eat cat food; it is too high in protein.
2. The dog should never be allowed to eat from the cat's litter box; even cat "waste" is too high in protein for a dog
3. Use a baby gate or some other obstacle and set up a place the cat can go and be in a "dog free" zone. Often, this is also where the cat's food bowl and litter box are. I have one spare bedroom gated off, and I keep the cellar stairs blocked from the dogs. The cats can come and go at will.

You will want to clip the cat's nails. A cat can unintentionally scratch the cornea of the dog's eye. This will be expensive, as it typically happens late at night requiring a trip to the ER vet (of course it does!). A scratched cornea will heal, but is very painful and the dog needs to be seen by the vet right away.

I am curious why you think your cat will run away -- is this an indoor / outdoor cat? Mine are indoor only cats, so running away was never the concern for me.