A bag of shredded mozzarella cheese was in my refrigerator. The door to the refrigerator was left partially open overnight. Can I still use this cheese or should I discard it?
Almost certainly the cheese is fine, but it may grow moldy a bit faster than it would have otherwise. Just use it sooner rather than later (within a couple of days if you can). Check it for mold before each use, and throw away the whole bag if it shows any sign of mold or otherwise being "off".
This study shows that nearly one quarter of all refrigerators operate at an average temperature which exceeds [edited] the optimum average temperature of 5°C or 41°F (the FDA recommends 4°C or 40°F). So even with the door of your frig opened and closed the proper percentage(s) of the time, already there's a risk that the temperature of your cheese was less than optimal for normal preservation.
Bottom line is this though. The USDA says all soft cheeses, including mozzarella (by its definition), should be discarded any time it's spent two or more hours (same link) above 40°F. So bearing in mind what's revealed of refrigerator temps above (first sentence), there's simply no question that you're taking a risk if you think to consume the product you describe.
It could, but the preshredded stuff is usually low moisture to start with. That cuts spoilage rate. Besides, the dairy aisle coolers from which the stuff is sold are often effectively refrigerators with the door left open themselves.
Cheese sits out for years.
I may think that the processed cheese may go bad, but never over night. It is crazy to even think that. You could probably leave it out for a day or two and it still would be good. I have left it out for that long many times and never got sick.
Cheese was left out for years hanging from the ceiling before refrigerators were even thought of, so go ahead and eat it with your crackers or by itself - my thoughts.