If you allow your dog on the lawn during winter, it will always wreck it, whatever you do - best to keep him off during the winter months and don't let him pee on the grass either - that burns the grass and it dies in patches. If you have lots of hollows, you can fill them with topsoil and level off, then sow grass seed on the area. Time consuming and you'll need to keep the dog off the seeded areas (and people) for a couple of months until its well established, but it's the cheapest way of addressing the problem.

I hate to say this but the only way to have an immaculate lawn is to get rid of the dog. As Bamboo has already said - if the dog goes on the lawn then it WILL dig and scratch. My personal choice would be to keep the dog and play on the "lawn" (which will not be a lawn for long).