I would make sure you have a tenant who is either employed or on the way to employment very quickly and who is rooted in the community. 

Another has had the electrical wires in the house 'recycled' to the extent that the electrical meter and main connection wire was removed to prevent a fire if turned on by accident or bypassed by the squatters. 

They could take it off the wall, which would prevent the system from starting at all, but they wouldn't be able to alter the temperatures or other settings.

Are there any rules preventing me from asking about their pet?

On Airbnb I create a new account for them (or update theirs if they already have one) and add myself as a cohost giving my primary Airbnb account access and preventing multiple logins.

Your city should have enforcement rules that require him to maintain his property in a manner that prevents damage to your property.If the city doesn't help, then you can try paying for it yourself and sending him the bill.

Onsite weekly visits to open the faucets and expel the gasses, adding the necessary chemicals to my drains to prevent oxidation, handle the bills, proper property maintenance.

To focus on having less of bills for a potential loss of income is like getting a set of denture made just in case ones teeth rot out, vs brushing them (skills maintenance) to prevent the loss in the first place.  

She should have been using rinse aid for the dishwasher to prevent Water spots or a better brand dish washer detergent.You need to nip all this in the bud.When my tenants ask for an upgrade I ask how much extra rent is this worth to you?

Now, let's tackle it like a pro: Identify the moisture source causing the mold, fix it to prevent a comeback.