How can I repair scratches in engineered hardwood floors?
I hate to be the one to tell you this, but there is really no magic putty or filler for shallow scratches and gouges in hardwood flooring. Even though there are a lot of colored wood fillers on the market, they are not designed for high traffic flooring or nuclear dogs. If it was just a couple of scratches you might be able to sneak by with a filler, but not for a refinishing job in a larger area as you describe.  Most engineered flooring has 3/16" to 1/4" of hardwood veneer over a plywood base. If the scratches have not gone through the veneer, then you can sand the scratches smooth, stain to match, and refinish with 3 to 4 coats of urethane. I would recommend an oil based urethane. Apply 3 to 4 thin coats rather than 1 or 2 thick coats. Remember, the veneer is fairly thin and usually can only be sanded and refinished 2 or 3 times at best. Once the scratching penetrates the plywood base, your floor is unrepairable.   

My advise would be to protect the floors with area rugs, runners, etc or keep the dog's nails cut short, or better yet, train them to behave in the house.