Bertsirkin:
We really wanted to go to Mitzpe Ramon and the crater, but just didn't have the time - Israel may be a small country, but there's a LOT to see!
That was my point : your itinerary is inefficient. Of course you don't have time for Mitzpe Ramon, etc, if you spend four nights in Eilat and three nights at the Dead Sea ! Very few tourists come for three weeks, yet they manage to see more in two weeks. There are no "locals" at the Dead Sea. Are you planning to visit Akko and Rosh Hanikra ?
There is no bus from the airport to Jerusalem, even during the week. You will have to take a taxi (about 300 NIS + luggage fee) or a sherut (62 NIS per person).