I can understand the no pet/smoking criteria.  

And then he gets a cat as a pet...wouldn't I be fully within my rights to fulfill the terms of the contract and keep his deposit?  

I am making offer on this duplex that has extremely strong pet odors everywhere, not only from the carpet, the carpets were already removed, I just think it's everywhere.

Huge pet peeve of mine when things don't function. 

One example: I don't think the lawyer would have a no pet or no smoking policy in his lease forms where mine does...

A few things to consider: Will you accept pets?

Find out why seller is selling, ask for a rent roll, what repairs or updating of mechanicals and roof has been done, how long seller has owned the property, are there pets allowed, are taxes current, has there been any insurance claims where seller did not use money for the repairs, ask for sellers disclosure, are there separate water and electric meters, does rent include utilities or not, are there long term leases or month to month, does deposits transfer to new owner, if there is pets, is the deposit a fee or deposit.

While I've found many references to looking for 'x' number of years of rental history in other posts, I can't find anything that directly answers this nuanced question:We have a couple that meets our income, criminal (lack thereof), number of occupants, pet policy, etc.  

We are two young professionals in our 20s, do not smoke, drink, have no children, pets, etc. and rarely have someone come over.

Unless it's general yard maintenance (such as picking up after a pet) I do not make anything the tenants responsibility for maintenance.