Well, it is a long-term guest, so still a guest (although he might have tenants rights depending on what the laws are where @Traci120 lives). I've had many guests who stay three months or more.
@Traci120most of the things you mention are annoying, but if they are not specifically against your rules, then they are just that. Yes, going through the trash is weird, but maybe he's just really keen on recycling and using his bike instead of the car in order to be more environmentally friendly.
What is absolutely NOT okay is how he treated your dog. Have you spoken to the guest about this and told him he can't do that? If a guest hit one of my cats, I would be straight on the phone to Airbnb to have them removed from my property. I am sure abuse of pets is a valid excuse to have a guest's stay cancelled.