I hate to bring this up but be very very careful with Lucy and the kitties. The first dog in my life was a beautiful, brilliant Shepherd mix who had been a stray. She was convinced that anything that wasn't a dog deserved to be killed, and was quite serious about it . Cats (squirrels, etc.) were either fast enough to escape to a tree to climb, or dead. My older sister was allergic to cats, so my parents had given theirs away, and Sheba was wonderful with children, people, etc., just not other animals. So I would be extraordinarily cautious in your situation, just in case they can never be happy together.
The neighbors, by the way, loved Sheba. There were no woodchucks or raccoons raiding their gardens until years and years after her death - that is how thoroughly she rid the "neighborhood" of "vermin."