Yep that is a fuel pump. Thats why I was wondering how much gas you had. The humming sound comes from the right because thats where the pump is. You are talking about the oil pressure light right? That is because the car losses pressure when it starves it self and kills the engine thus no oil psi. When you floor it the pump tries to suck more in real hard stopping it from stalling but causing it to jerk.
I bet people like Benzmanc can tell you more.