Management company wants to keep pet fee, I want to claim the pet fee and be responsible for any damage.

We are priced about the same as the apartments, and while we don't have a gym or pool, we do have a yard and accept pets. 

We’ve had an apartment sit vacant since August and have had no luck finding decent enough tenants that don’t smoke or have pets, have no felonies and no prior evictions.

I guess I could call one of their past landlords and see what damage his pets did to the house and see if they have any ideas.

Check the furnace filter, make sure downspouts are attached/down, ensure tenant is caring for landscaping, check on smoke detectors, are there evidence of pests, does it look like more people are living here then indicated on your lease, and finally is there evidence of pets not on the lease. 

my tenant moved in on 11/1/2019. she is on a 1 yr least. before her lease signing I had told her a pet (small dog or cat) would be allowed with a $25 per month pet rent added to the rent.

She paid a 250 pet deposit, 975 deposit and first months rent.

(no pets allowed got overlooked).

How many pets do you have?

In addition, ask about important factors like pets, children, smoking, and criminal history