I'd also go prepared with info on actual recently sold comps in the area then gently go over those numbers with them.

The termite and moisture inspection didn't even mention that particular floor, and I had another mold remediation expert come in for a full look-through, and he didn't note any mold there either.

His recommendation to me is prior to renting your unit out, get documentation of moisture readings on the floors, walls, and air.

Gently, of course, but I tell them I've got crews and employees and they do all the work for me.  

Or other similar bristle brushes would work.I would caution that if it's hard duct pipe may be screws to hold the sections together that might get hung up if you tried to run a towel through, but I have done that with success - gently pulling of course.If it's flexible duct pipe, well, they do come disconnected quite easily.Then there is the vacuum hose if you have one long enough and can safely access the roof with it - then access from the inside as well.Hope that helps some.- Joe

I am assuming I need to hire a water leak detection company to use some sort of moisture detection equipment.

The joist has swollen from all the constant moisture. 

It was built in the 1960's and apparently the moisture has caused the flooring to rot and has given way.  

If not, then I would say-- gently, but I would say it anyway, that the asset is a vehicle for investment purposes.  

No visible water, probably just moisture coming in from the foundation.