Anytime the fuel pump stays running with the key on it's either a stuck fuel pump relay or most likely a bad engine computer (PCM). Combined with the other issues you stated, it's most likely the PCM. Remove the connector, the retainer screws, then pull it out and open it. Look for signs of moisture, burn marks or leaking capacitors.

The PCM is located under the brake master cylinder/vacuum booster. It is removed from the engine side of the firewall.