My theory is maybe an installation or moisture issue, but I really have no idea.

I will have paid for all of it as the repair and getting rid of water/moisture was an asap thing so the tenants could go back to their lives. 

The seller also noted that any repairs needed to any correct any issues on the moisture report would not be paid for by them.

First you need to talk to the seller about getting an HVAC Tech, or handyman, to close the service valves on the condenser, with an allen wrench or service tool, so that moisture, non condensibles and bugs don’t enter the unit.How close is the unit to the house?

I would not leave it solely up to the tenant to deal with it...... but you need to attack the source of the issues which is often tenant hygiene...... mice come and stay where there are food sources, moisture and warmth.....

A fogger hits all of the surfaces quickly, dries quickly because its not a moisture saturating fog and you can then close up the walls once you confirm the smell is gone.

I want to highlight at the lease that tenants should monitor moisture to avoid mildew growth and clean if they see it, they should change air filters quarterly, smoke detector batteries are their responsibility to change, etc.

In most cases, the tenant caused the mold by not keeping the unit clean, or by not ventilating the unit to reduce moisture.

I imagine if I eggshell or satin sheen over it, it might seal the wall paper and keep it from the moisture of the drywall mud.

There was moisture damage to many walls and some ceilings as well.