The paper notes that a 1-in-1000 molar concentration of acid was sufficient to trigger low-temperature melting, indicating that the amount of contamination needed is indeed very, very small. The paper does not discuss exact mechanisms of action, but one of the conjectures is that the acid takes the form of excess hydrogen atoms, which could easily tuck themselves into a crystal structure nearly undetectably.

All it takes is catalyzing a few Z->E transitions when the temperature rises to cause the crystal structure to break down, which would explain the lower melting point despite the small amount of contaminant (catalysts often reduce the energy required to start a reaction, but are not themselves consumed so they can further catalyze other reactions).

Crystals are repeating arrangements of atoms bonded in a very particular, repeating way. Modern techniques include looking at the crystal with x-rays, so any serious discrepancies in the patterns would be noticed.

The contamination, they concluded, was not structural (e.g. as a silicon semiconductor might have random atoms replaced with phosphorus or boron) but rather even more minute amounts of acid could act as a catalyst - i.e. an extra component of a reaction that participates, but doesn't actually get consumed - hence can make a difference in tiny amounts. A decent way to think of catalysts is as chemistry's matchmakers - they help reactions happen; sometimes those would happen on their own but take more time, or sometimes they wouldn't happen at all (e.g. if there's not enough energy to react without the shortcut of the catalyst).

Hope that makes some sense.