If there is a lift pump, if you go into your boot and lift the carpet up you should see the top of it or at least a round metal plate with some wires going down to the lift pump. If you disconnect the pump from under the car, as in temporarily pull the wires, turn on ignition and listen you should hear the lift pump prime, if you don't hear anything it could be a contributing factor. However it doesn't explain the fuel pump constantly priming? I'd be more inclined to be looking at the electrics of the car, first relay, then the dodgy immobiliser.