If they feel like their is mold although someone has already inspected for that ... you can get one of those water scanner things (sorry I have no clue what it is called) but it essentially checks for moisture (it shows certain color based on the moisture) also I would remove the base boards and cut about a foot of the dry wall to see if that is wet.

Address the water/moisture problem first, then remediation.  

Add in water/moisture sensors, etc and you could have a fully-wired house sending you alarms to allow early intervention to what could become major issues.

(from our core rental agreement)Mold may grow and exist in any structure where there is, or ever has been, a presence of moisture and a food source for mold to grow.Its presence may exist without the knowledge of the structure owner and may be concealed from the untrained observer.Some varieties of mold are toxic and may cause adverse reactions in certain individuals.Tenant and Landlord will complete a Mold and Moisture Hazards addendum that will become part of this rental agreement.MOISTURE AND MOLD.

I’d rather trust a professional with a moisture meter to see if it’s dry (just after or even during a heavy rain would make me feel better) before fixing the ceiling.

The first part of Mold remediating is stopping the moisture issue  until you figure out that part don't bother with step 2 . 

He’s put in tile and LVP floors so no carpet to clean or replace, the bathroom lights are wired with automatic fan turn-on which prevents moisture accumulation resulting in mold, etc.  

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 care for the premises, keeping them clean and clear of debris.""

If the tenant identifies an area of moisture that may be causing growth we immediately address the source of the moisture, replace and damage building material (usually dry wall) and or use mold cleaning solutions/mold blocking primers as required. good luck

HVAC duct work insulation is also deteriorated.No vapor barriers installed eitherSome fungi/mold growth notedPlumbingMultiple small leaks under sinks and from bathtub and toilets.House has Polybutylene plastic plumbing which is dangerousMain shut off valve is damagedElecticalSome small exposed wiring and bad splicing.Whole house has no GFCI's or grounding outletsInteriorMoisture spots on ceilingExteriorLose Vinyl SidingDoor Jams and Window trim have moisture damageDriveway need replacementRoofingMinor leaks noted in atticRecommended to replace roof covering, but not whole roofSome fungi growth noted in roofSo some of the question I am trying to tackle with this work is:How much should I budget for the main issues listed?