I still need to play with the camera settings and go back for more photos, but, you can see below a reminder of why we have a No Pet policy:This is what many years of "doggie diapers" at the end of the hallway look like ... yes, the yellowish and brown areas are urine stains.  

I would decline to do any cleaning  Yes, the tenant may have stained the carpet already and is preemptively trying to get out of a future damage charge.

Tile, wood-look waterproof vinyl planks, stain concrete, etc. will be your best friends.

I believe I cleaned all living spaces as well as one could, minus normal wear (nail/tack holes in walls, nicks, light staining etc).- I cleaned kitchen and bathroom NOT QUITE AS THOROUGHLY AS I WOULD'VE LIKED.

@Ryan SwanLuxury vinyl plank has become quite popular in recent years due to its resistance to water and staining.

They can also purchase a humidifier, open the windows, or take other steps to prevent growth.The mold will cause staining, if it hasn't already.

for this reason I will never directly work with any wholesaler save a very very few... that personally know well.its a stain on our profession these unlicensed people playing investor.. :)  

What is the best way to cover up or remove the stains?

I've had to paint one bathroom with water stains on it and my tenant has never asked me for anything since.

One trick I have used to get the scuffs out is a color match stain stick.