I think that you are spending too much time in Eilat (unless you are a diving fan) and the Dead Sea. Eilat is just a standard beach resort and you don't need more than 2 days at the Dead Sea: Massada, Ein Gedi, Qumran and a spa do not require more.

I would rent the car in Tel-Aviv and drive down via Mitzpe Ramon, stopping at Bet Guvrin, Sde Boker, Avdat, the Ramon Crater, Timna on the way. I would also add two nights in the Haifa area and tour Akko, Rosh Hanikra, Haifa, Zikhron Yaakov and Caesarea.

Don't exaggerate Jerusalem traffic; I don't like driving in the city either but you only have to get to your hotel and then return the car. It is a minor inconvenience (and much cheaper) compared to returning the car at the airport. What hotel are you staying at ? They can give you detailed driving instructions or you can ask for directions here.