If you store an open package of grated parmesan cheese in cupboard will it get stronger in flavor with time?
No, it will actually lose flavor...I know, I've tried it.
You should leave cheese aging to professionals. Leaving the container open probably won't turn out well. The cheese would either dry out or absorb unwanted moisture. You also risk the slight chance of it molding, and that would ruin the cheese.
no, since the condition in the cupboard may not be proper for aging. it will only lead to contamination of microbes, and eventually, spoilage.
No, it will go off, and even lose its flavor. Best place to keep it is the frig.