In addition, we rerouted the airways to reduce moisture and make it more efficient.

"Gently used bungalow, with dedicated garden, rose bushes, privacy fence, natural greenery (and Jorts)!"

Install something for moisture meter in basement. 

Our lease statesIn signing this Lease, Tenant has examined the Premises and certifies that Tenant has not observed mold, mildew or moisture within the Premises, unless otherwise noted during the move-in walkthrough. 

The tenant has not told about any water leak or moisture issue or mold till date.

Do keep in mind that you will have to do more than a gentle rehab for it to qualify - you have to put at least as much into the property (over a contig 31 month period, IIRC) as the purchase price of the property (excluding land) to count as 'substantially improved'.

Sounds like you are doing the right thing with the moisture management.

It's not going to matter unless you have some kind of weird vapor barrier preventing moisture movement below the sheathing - water vapor moves through the roof sheathing all the time, and wet wood will dry out the same way (water vapor moving from wood to drier air). 

Fold a few pieces of paper towel, dampen, and then gently press down on the pile of broken glass. 

My lender heard from the appraiser that there was moisture in the basement of a couple places I am trying to refi, and all hell broke loose.