@Sui201 Yes, they should. But Airbnb seems to turn a blind eye to this sort of thing particularly if the host has many listings. And whatever investigations Airbnb undertakes, they won't share that info with you.
It's really safer to stay away from listings where the host is a property manager with tons of listings, as these are the ones that seem to get the most complaints.
Re the cat. You say the host allowed it, even though the listing says no pets. So did you ask before you booked if you could bring the cat, or did you just show up with it?