Is it the oil pump.

Water pumps can make noise even though the temperature is normal. Oil pumps rarely make noise but if they're worn they can cause noise in the valve train first, and from damaged bearings later. The best test is to install a mechanical oil pressure gauge to see what the actual pressure is. If it's low, the place to start is by measuring the bearing clearances and inspecting them for wear and damage. If the clearances are within acceptable limits, THEN suspect the pump. You may be able to rent or borrow the gauge from an auto parts store that borrows tools.

Another fellow recently had low oil pressure with his Toyota and it ended up being a broken spring in the pressure relief valve in the pump body. He ordered a new pump, installed it, and still had low pressure. Turned out the cup that holds that spring in wasn't seated properly from the factory and oil was leaking past it. Who would have ever guessed that, but he found it, made the quick repair, and no engine damage resulted.