Your oil pressure sensor is probably gummed up. They do that occassionally. It's a matter of pulling oil pressure switch spraying or soaking in carb cleaner, drying it off and putting it back in. No big deal. For the most part you know the sensor is working.
As far as being connected to the fuel pump - YES it is. There's a safety feature built in (because the majority of our cars have an electric fuel pump in the gas tank) it's to prevent the fuel pump from pumping fuel if you roll or crash the car. If you were to crash or roll and the engine quits, the drop in oil pressure shuts the fuel pump off - so your hot engine (and yourself) doesn't get a bath in gas or worse a fire.