The interior side casing of my window has been scratched/chewed away a bit, destroying the paint and outer two layers of plywood:
What's the best way to repair this?
PS the source of the problem might be this guy:
Start by clearing out any loose particles that are still clinging to the wall.
Once you've got it cleaned up, wood filler will take care of the missing plywood patch. Apply it with a putty knife and sand it smooth after it dries. It will be easiest if you use a putty/taping knife wider than the patch you're repairing. Otherwise, you'll have to do a little more sanding to smooth down the ridges created between passes - either way, it's not a big deal. (If you've already got a smaller putty knife, I wouldn't go out and buy a big one.)
After sanding, wipe the area down to remove all dust, then apply a coat of primer.
After the primer dries, paint the patched area. Try to feather out the new paint overlapping the original. It looks like you've got plain white semi-gloss or high-gloss paint here. Don't forget to match sheen as well as color, unless you're lucky enough to have some of the original paint left over. One or two coats should be fine, especially if you primed beforehand. Also try to use the same method in which the paint was originally applied - it looks like a brush was used originally, so go with a brush for this. A roller will leave an 'orange peel' effect, which will stand out from the surrounding area.
PS: Don't bother repairing unless planning on trading in the dog ;-)
I've used Bondo (the automotive body filler) for this type of thing. It adheres better (in my experience) than wood putty, is stronger, and is similarly easy to sand. It also cures by an epoxy-like chemical reaction with a separate hardener, so it hardens quickly without cracking no matter how thickly you need to apply it.
It's typically available in box stores in a quart-or-so size which would be perfect for this.