Did you try shifting in to neutral before starting? Maybe the screw was for the Transmission Range Sensor and it's now out of alignment.
Do the indicators in the instrument cluster perform a bulb test in the Run position but go out when the key is in the start position? Are you familiar enough with the starting system and a test light to check for power to the solenoid when the key is in the Start position? Have you checked battery voltage? Attempting to jump start the car isn't likely to cause damage as long as the cables were connected properly.
-Rod