A pet addendum sets the terms on which an animal may live at the property, and there is one distinction that has to be understood before any of it: an assistance animal is not a pet under federal fair-housing law, and the rules in a pet addendum do not apply to one. This page is the addendum, the money, and that exception.
What the addendum should contain
Which animal, described specifically, because permission for a named animal is not permission for any future animal. Where it may be, any limits on leaving it alone, the requirement to clear up and to control noise, and responsibility for damage. And the process if it becomes a problem: what you would ask for, and in what order, rather than an immediate right to end the tenancy.
Pet money, and what it is
A pet deposit is deposit money, held and returned under the same statutory rules as any other deposit and applied only against actual damage. Pet rent is rent, which is yours as it accrues. A non-refundable pet fee is regulated or prohibited in some states. Which of the three you are charging should be stated unambiguously, because tenants routinely believe pet rent is refundable.
The exception: assistance animals
An assistance animal accommodating a disability is not a pet under the Fair Housing Act, and pet restrictions, pet deposits and pet rent do not apply to one. A no-pets policy does not permit refusing a reasonable accommodation, and there are limits on what you may ask for in support of a request. This is a point to get right rather than to improvise, and it applies regardless of what the addendum says.
Where the damage argument is won
The inventory. Photographs of the floors, the doors and the yard at move-in, dated and agreed, are what make a pet damage deduction stick, and their absence is what makes it a negotiation. Take them at every tenancy with an animal in it, and take them again at move-out before anything is cleaned or repaired.
Questions people ask about pet addendum
Can I charge a non-refundable pet fee?
It depends on the state and several restrict or prohibit it. What is never safe is treating a deposit as non-refundable by calling it a fee in the document.
Can I refuse a pet outright?
Generally yes for pets, subject to local rules. What you cannot do is apply a pet policy to an assistance animal accommodating a disability.
Does the pet addendum need signing by everyone?
By everyone on the lease, yes. An addendum signed by one of three tenants binds one of three.