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The drywall that has water damage should be removed. The wood studs might also need to be replaced depending on if there is mold or rot. The insulation behind the drywall should also be replaced. The base molding might also need replacing as well as the flooring.
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.
Is your garage attached to the house or is it on its own? If it is attached is there living space above the garage? Are you looking to insulate the walls and ceiling? Are you going to drywall the garage? I would suggest using batt insulation, with a vapor barrier. I would then drywall everything. Before you drywall I would make sure that you have all the electrical and anything else that you want in the walls installed.
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