I suspect moisture damage on the laminant in one room is from pee.

None of the problem areas are in exterior walls, so I don't think moisture is a factor.

I am taking down a few small trees on my property to get a little more sunlight on the decking and reduce the moisture, and I'm hoping the neighbor will remove the bigger one on their side before too long, since it's too close to their house and will have to go eventually.

Moisture in our environment is a real issue and although I haven't heard of this situation, your uncle's explanation makes sense to me.  

The inspector said there was only minor things that need to be fixed (like curvature in the bathroom wall due to moisture hitting it, main door warn out, appliances are really old, tiles are cracked, uneven paint job, moisture in the cabinet below the sink, some electrical units wiring needs to be fixed as the hot is on the right, etc etc), but no major problems. 

It's pretty rough here.....There was so much moisture in the crawl space that the subfloor warped.

Tenant agrees to allow Landlord timely access to premises to do maintenance and repair.Tenant will notify Landlord in a timely manner of any damages or problems as they arise, including damages caused by Tenant, malfunction of appliances or fixtures, and water/moisture leaks and spots.Tenant will take care of the premises, keeping them clean and clear of debris.

I've tried to gently prod him to consider what may come up, but alas he does not quite get it.

In the end the truth will come out so no need to mislead them from the start, they will appreciate the honesty.With cold calling I have found that a soft and gentle approach quickly extinguishes an irritated call recipient. 

I smelled bs and I did it anyway and we found mold in the basement ceiling rafters, caused by bad gutters allowing moisture into the basement.