The inlet to the fuel pump, in the gas tank, won’t pick up a piece of ice because there is a filter over the fuel pump intake.
If the oil pressure switch opens, it cuts power to the fuel pump. Loss of oil pressure would cause the oil pressure switch to open, AND CAUSE THE OIL PRESSURE WARNING LIGHT TO LIGHT ON THE DASH.
The engine starts, and runs, for 30 seconds, and then dies? If it still does that, it could indicate that the oil pressure switch isn’t powering the fuel pump. If the low oil pressure light doesn’t come ON before the engine dies, the oil pressure switch could be faulty.