Just a wrap up: The most important clue was the need to have the car jump started only after a few minutes after shutting the car off. All cars these days are fuel injected (a computer-controlled system that delivers the fuel in measured amounts to each individual cylinder) but for the injection system to work while starting the car the battery has to deliver at least 10.5 volts while running that incredibly power-hungry starter too. A weak battery just can't do all of this at once!