you are describing would happen if the battery wasn't fully charged or had gone defective. Voltage would drop significantly when any load was applied and the car would be hard to start via a jumper. I realize the battery was tested but the symptoms described do point that way....
Have you tried substituting another battery - could be from another car. Jumping the car w/out the old battery connected would be another way to test -- connections to the car from the other battery would need to be well made to insure enough current was supplied for the starter to go.