I grew up on a farm in northern MI and we had dogs and cats, both, inside. When the cats were first introduced to the dogs it was almost hilarious. Dogs will only chase a cat if it runs, and in a house, there isn't enough room for that, so the cat finds a place high enough, and then sits and hisses and growls until the dog tires of the game. Eventually the cats and dogs learn to accept one another and will actually play together, especially if the cat started as a kitten.
But I digress. I have never tried this, but I have read it more than once. Mice don't like the smell of peppermint. We used to make our own hard candy for Christmas and we bought peppermint oil at our local Pharmacy. This article suggested using peppermint oil on cotton balls in your cabinets and changing them frequently. It won't solve all your problems, but it might keep them off your dishes. I think that Steve's suggestion of a combination of carefully concealed poisons and traps and an inside cat is probably your best bet.