Hi Joe, sounds like the pump is ok as the pump operating for a second after shut down is what its supposed to do to take up any air in the system and if its only running for the same time on start up thats ideal, which brings me back to the rail pressure regulator which is a spring loaded device that is solenoid controlled but it can have a tendency to stick when worn as they are not individually monitored by the ECU they generally throw up a code for rail pressure the clue could be that the car only does it on rough roads, they don't like being rattled around when worn, the reason for the effect not happening when accelerating hard is because it will use a part of the valve stem in the unit that is infrequently used, once the valve is worn they can stick and starve the rail of pressure as it is slow on reacting to the opening pulse.
Steve.