I thought this would help offset the decrease in value of the property caused by any reduction in rent.

Cause then she could be using some loophole and you'll be caught unaware.

Just trying to understand why someone would choose one over the other or what type of scenario would cause someone to do so.

FWIW, we have purchased property with tenants in place paying "pet rent" and never received any additional funds form the previous landlord- there was no account sitting and collecting the "pet rent" that would be used towards fixing any damages caused by the pet.For this reason, we don't charge pet rent or a nonrefundable deposit- it gives the tenant the impression that any damage is covered. 

What is a "friendly" request, could end up being a cause of relationship friction down the line.  

Any evidence of pets on furniture may incur extra cleaning fees.Pet must not cause damage to premises or furnishings.

According to MLS it's a probate sale and the owner died in the property in February of this year of natural causes.

If their personal credit score is poor, you can be assured when something causes less than favorable returns, you might not see a return.

Cause honestly most realtors are not going to those kind of offers.

The tenant says the black on the walls is from the fireplace flue not being closed causing smoke to get on everything.