., pets).How should the contract be handled moving forward?

If pets are allowed by the HOA I would strongly recommend you consider allowing pets as more than 50% of travelers travel with their pets and that will make your priority more desirable.

We do NOT sign a lease with a prospect until they have:1) Paid Holding Fee2) Paid all MoveIn Funds: first month's rent + security deposit + Pet Fees, etc., MINUS Holding Fee paid.3) Utility Transfer: we require proof they've put all applicable utilities in their name.4) We do send them a copy of the lease to review.Once all above done, we send lease to sign electronically.

Consider 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.

First, you are making a mistake keeping the tenant who violated your no-pet policy and hasn't paid their share of the rent ($50 a month).

It would be good if traveling with pets required number and type but that is just nice to have. 

Call DEC and tell them they are harboring a pet squirrel - that should be enough to send out a crew of people to get rid of them.

You can charge pet rent, a pet deposit, or both.

It's also good for insurance and legal reasons, because you will need to indicate every adult and all persons, as well as details (like children, pets, etc.), on the lease.  

Consider 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.