You said she's managing the facility. Does that mean The original owner still owns it? If so, I'd call your former boss and voice your concerns.

The way the dishes are getting washed seems kind of picky. She washes them by hand rather than in the dishwasher, so what? They likely get cleaner that way. But the rest are definitely legitimate concerns. I think it's important for these things to be brought to the attention of the owner. After all, she's the one who will have to answer (and pay) if her kitty guests start getting sick from lack of proper sanitation. It's her reputation on the line as a business owner and I'm sure she wants her business to be represented as professionally as possible.