We can easily show the stained carpet.

There is certainly a stain in that picture, but the photo itself does not appear to reflect mold. 

When when moved out, the carpet downstairs was full of ink stains.  

Next thing you know, you've got a greasy spreading stain.  

And she also sheds like crazy, and will occasionally barf out of the blue on the carpet, and every time I bring her in and out of the apartment, she simply refuses to wipe her dirty feet :-)In other words, even really great dogs will still shed and puke and get sick or have accidents and bring in dirt that might stain.  

We’ve had dark stains all over.

Ex: if I buy a sander to address stains from tenants on a wood floor can I deduct the cost of belt sander from the deposit?

All damage/stains/odors would be the responsibility of the tenant to correct/remediate/repair. 5.Damages / remediation efforts arising from pets in the home would come from the original deposit and not the Annual Pet Fee or the additional monthly charge. 

But a huge stain or damage to the carpet would be damages.Assuming the only issue was they left the condo "messy" then $775 seems excessive to clean it.

The tenant attempted to clean the oil stain using a commercial cleaner, but was not successful.