That grass looks pretty and delicate. Have you thought of getting some
st augustine grass? Its like little vines that grow and form a nice thick pad of lawn, real nice stuff. It is what we have all around our property but down here it grows like crazy. When I was a kid we had a lawn of st aug grass in So Cal and it was super nice.
It really looks like you'll either have to put in some stepping stones or try a tougher turf.
...or get rid of the dog.
