It sounds like they charge a tenant whatever fee they can get and that is not a pet deposit in your favor. 

I'll open a local bank account and have them all deposit their rent automatically or pay online.

Then you just bought a whole lot of trouble with your assignee.You might be able to limit your liability to your deposit, but that's an obligation.

Only been sued for return of security deposit and never lost. 

LOTS of people have pets.We simply use a one-time pet deposit fee (non-refundable) and limit it to a cat or a less-than-25 pound dog.We've not had an issue yet.We make them sign a Pet Addendum which lays out requirements like carpet cleaning  and flea and tick treatment prior to move out.Best of luck!

Should I let them go and just keep their deposit?

You have all the control.You could stay in your own Vacation rental for free for a period of time (add this to your agreement)Make sure of their abilities, don't rent to someone that this is their first attempt at a STR.You have little risk in this scenario.Get first, last months and a security deposit to further minimize your risk and screen them like you would a normal tenant.

Think of all the things that could go wrong and see if the lease addresses them: unauthorized pets or tenants, early termination, security deposit, lease violations, late rent, eviction, lawn maintenance, parking, etc.5.

* What do you do to ensure that the tenant is responsible for security deposit disputes since that is the largest reason for owner lawsuits* How familiar are you with the newly changed laws that can affect you the owner if they are not used correctly?

with rentals cutting grass seems to be a issue ,I always press the issue when renting out a house .call the city on your self then post the paper work  on their door .Then hire it out the mowing , raise the rent to cover the mowing when the lease is up  and keep the deposit if they refuse to play nice.How a person keeps the outside is a clue how the inside is kept be on high alert !