Look into Idle Air Control Valve (IACV) there are threads on cleaning it and this might help the bouncing/hunting idle
Find it (on Intake Manifold near firewall).....disconnect the wire plug, the bolts holding it to the Intake Manifold and the two coolant lines. If it looks dirty....Clean it with some Carb cleaner thoroughly (maybe 2-3 times) and let it dry well before re-installing it
Not sure on the fuel pump being bad but it might be. You can also listen when you put the key to the "ON" position. You should hear the fuel pump click (noise from rear of car) and the main relay should make a click or two (under the dash....somewhere...not sure about your DA). See if it does it when it starts and THEN see if it does/doesn't do it when it dies on you