We got the cork backed fancy NuCore with 6-mil top layer (waterproof and lifetime warranty for residential and 15 year for commercial) for $2.89/square ft -- that just needs a moisture underlayment @$.11/ square ft over a concrete subfloor and doesn't need anything over tile or any other kind of subfloor. 

But, if mold is the result from them not using the effective bathroom fans that are in place, they will be in violation of the lease and will be held accountable for any moisture damage that results, such as mold.3) they saw the ceiling fans when they looked at the unit.  

Could be because of something with the downstairs shower and the moisture rising?

And check the moisture situation.I don't know about hiring a gardner.  

The tenant believes moisture is getting in from under the unit, however, we believe this is from their lifestyle. 

From the underside you can see there as been moisture beforeThis is our first rodeo so after getting the builders opinion and servpro we went ahead and called our insurance.

Would not running air conditioning setup conditions for moisture and mold to develop in the unit during the heat of the summer? 

Try to identify the smell... grease, pungent food odor, mold/moisture, tobacco/other smoke, plastic off-gassing, paint off-gassing, pet smell, rodent odor, etc.

You can get a moisture alarm that will go off when a sump pump fails.

Do some reading on basic HVAC design and you will learn that a larger system doesnt work better due to the fact that is does not run long enough to adequately cool the air and remove the moisture that makes you feel hot.