What are common causes of smelly HVAC?
The stench is certainly&mdash;due to your hot-weather-only mention&mdash;due to condensate dripping where it shouldn't.  While the majority seems to make it into the drip pan where it is safely pumped away, some condensate may splatter into the filter, plenum, and/or ducts where it activates dormant bacteria.  Also the "return" refrigerant line may well be cold enough to create condensate on its surface and drip to bad places.

I would start by taking a good look around the evaporator and air filter.  Temporarily remove the filter and see (smell) if the stench ceases.  Wipe down all surfaces around there (while the air handler is "off").  Maybe even spray everything with a 100% white vinegar or 5% bleach solution.

Also look at the evaporator itself:  is it partially blocked?  Does it look clean?  What about the edges which don't "work" as hard in not-so-hot weather?  Cleaning it requires great care not to block it further (by fin folding), but a hand-held spray bottle of water can gently work wonders in many cases.

As for earplugs in the ducts:  My wife sometimes uses them to get to sleep; they easily fall out during the night and travel surprisingly far by morning&mdash;either onto the floor or between the sheets toward the feet.  If there is a floor register within a few feet of the bed, I would not be surprised if they rolled right on down there.