Answers

Consider, a new dog

It looks like the dog did a pretty good job on the door. Since the door is painted, you could sand the area down with some 150 grit sand paper. After it is sanded then you can apply some High performance wood filler or car bondo.

Apply a thin layer over the damaged area and allow it dry ( a few minutes or more depending on the amount of hardener you use). After it is dry, sand untill smooth and then apply a second layer if you need to and then sand smooth. After the area is the way you want it, prime it and paint it. 

There are a few options depending on the circumstances, if the dog has scratched the slab and depending on material, you'll need an entirely new slab. You have the option of painting over the scratch, but the odds of that peeling into the future are high. Our doors are steel and fibreglass, fibreglass would require an entirely new door slab to be installed if you wish to rid the problem for good. To surely rid the problem, the dog or puppy should be trained before repair unless you want to experience a fruitness endeavour. 

When dealing with a wooden door, their are a few DIY solutions like sanding, filling sanding again and painting/staining over. 

It looks like the pup teaches classes on how to be a beaver 101. Sanding, filling and matching the colour back would be the best option in this circumstance, or simply an entirely new replacement.