Regarding losing prime, mine does the same, but I don't know how quickly. If you fuel pressure regulator works while the car is running, than it should bypass only enough fuel to reach the target pressure after shutting down. Most likely it's the fuel pump which should have a check valve. Mine takes only one key on cycle and probably three seconds to reprime so you must have something strange going on. Perhaps a leaking injector or bad regulator which could be causing your timing issue. Most EFI systems use timing as the primary means to control the idle speed and if you watch your diagnostics, the timing varies greatly even when the car is idling as it should. Perhaps a fuel problem was causing the timing to retard or advance too much in order to control timing.