Dogs use their mouths and paws as hands. I give my dogs snack "sticks" wrapped in bacon; they all lie down, hold the stick between their paws, and use their mouth to carefully peel off the bacon to eat it first. The stick itself is crunchy but edible; they do the same thing: clamp it in their front paws and use their molars to bite pieces off.
This is not just one dog, but many dogs. Just like an elephant can learn to use its trunk to hold a paintbrush and use it, or to wield a hammer, a dog with human intelligence and ingenuity would devise tools it can wield with its mouth or lips (which, as you've seen when they bare their teeth, are movable).
It would develop "lying down" desks (think of an inclined plane) to keep it's rear end near the ground but its chest raised but supported, so its arms and paws are free to move, and it can bend its head to work on things on the bench. For finer work, it would invent tools it can operate with it's mouth that can move or cut more finely; just like we invented chisels and needles and writing instruments that work more finely than our fingers and nails. Then tools are used to create even finer tools, lenses, microscopes, telescopes. Tool making is a bootstrap function, we began with sticks and stones, shells and rocks. Dogs would do the same.
The important thing in human development was never hands, it was abstract intelligence. Elephants and dolphins both invent and use tools without any hands, crows shape tools shaped like saws out of stiff leaves, cutting them with their beaks. The dolphins tear out and shape sponges to use like a glove on their snout so they can stir up dirt to make crustaceans jump, for a treat. They'll carry these gloves a long distance to use them.
One of my dogs understood trade! If I were eating a snack, she would bring me one of her toys, drop it at my feet, and look up at me to see if I'd take the trade. (Of course I did.)
Humans with just arms and no hands would figure out how to survive and make tools they could use with their arms and feet; they wouldn't just lay down and die. The fine control of fingers is great, but in the end our tools make that control even finer, even tens of thousands of times finer. It is a compounding process, every time we make a tool (like a lever) that can convert a large action into a more forceful small action, we can use that to make more tools that use that small action to make an even smaller action.
Thinking in that direction, the tools of dogs could be lever controlled: A pivot well off center, so we have a long "paw arm" and a small "action" arm. The dog moves the paw arm five inches, and the action arm moves 5/16ths of an inch. Stuff like that.
Big machines with large movements to make small devices with small units. The first "machines" we developed were likely levers and fulcrums; dogs could certainly manage that, too, using their mouths and paws.
