Is the building brick or frame stucco.Look for damage at all door and window jams and around any source of moisture.

She’s the sweetest most gentle dog.

That white stuff that looks like mold is actually called effervescence, which is the block sort of disintegrating from the moisture content.

Second have a non confrontational meeting with your tenants and gently explain to them what you have learned.  

Hardwood is beautiful but expensive and easy to scratch and is finicky around moisture and humidity. 

Is there a reliable method/way to maintain a constant moisture level year-round so as to protect the foundation from shifting that would compromise all of the extensive renovations that have taken place?

Any moisture trapped will just cause mold.If the cracks are small or hairline cracks you can most likely go over the existing grout with no issues.Where the tub meets the wall there should be no grout EVER.

Shop around for gently used or scratch and dent to reduce costs. 

Low spot is about 1/2" lower than a high spot.We also found that the puddles form in the crawlspace when it rains, and it's very likely the flooring sagging due to structural damage from the moisture.

About $6K to replace the HVAC itself.Give the mold, you might eventually need the air ducts cleaned.Do the bathrooms have vents to get rid of excess moisture during showers/baths?