I will be coming from NYC and will not have a car. I am open to renting a car if it is necessary. Thank you Shuffaluff, and Gmar Chatimah Tova!
Hoangers - you are trying to do way too much and, IMHO setting yourself up for a miserable experience. You have to better understand both the significance and scale of what you are trying to see and the distances between them.
Jerusalem is one of the world's most fascinating cities. I spent over a week there last year and barely scratched the surface. When you include the surrounding area (like Bethlehem) there's even more to experience. Petra is an entirely different experience. Petra is geographically large and very verticle, meaning you have to climb up and down hundreds of stone steps. It would take you two full days to see most of it. Wadi Rum is an expanse of desert requiring one full day and night. Now, consider it takes a full day of travel just to get between Jerusalem and Petra and there's simply no way you can do any of it justice in only seven days.
What is your highest priority? If it's religious in nature then spend your week in and around Jerusalem. If it's nature then go directly south and see Petra and Wadi Rum. You simply can't do it all in just a week when you'll consume three of those days in transit.
Good luck.

Hi Laketraveller
Thank you for your insight. I agree that it is a very ambitious itinerary and like you mentioned, there is a very good chance that I will be busy rushing through things that I will not have time to take in the full experience of being there. My priorities at this point are 1) Jerusalem, 2) The Dead Sea, and 3) Petra. I should probably leave out Wadi Rum.
Do you know if it feasible for me to leave Jerusalem in the AM, rent a car and drive to the Dead Sea, spend a few hours there, then drive to Eilat and return the car there? Or is it better just to take public transportation. How long should I anticipate to cross the border over to Jordan?
Thanks again.
While the answer is based on personal opinion, I would take the bus. First, with exception of Friday afternoon and Saturday, public buses connecting Jerusalem, Eilat and points between are frequent and relatively inexpensive. Second, renting the car involves more than just the car. You also have to consider parking, gas and time spent getting it and turning it in.
Last, driving in Israel is basically fine, but driving habits are different in different countries and you might find driving in Israel a bit stressful.
Have fun.

Hi, if you ask me, the one place which has the most memorizing atmosphere in Jerusalem is the Holy Sepulchre, i guess it is on you'r to do list there, but if by any case it isn't well.. it should be :)

Hi Mefager,
Thank you for your recommendation. I am very much looking forward to visiting The Old City in Jerusalem. Are there anything else in Jerusalem that you think that I should see?

Israel + Jordan in seven days does not seem realistic. You will need two days to travel to and from Petra, and at least one full day to see the highlights of the site (you need two full days to really see the site). Jerusalem itself is worth two days minimum. You will be rushing everywhere.

Hi Catw,
I agree that it will be rushed. I have revised my itinerary to the following. I would be interested in what you think.
SUNDAY
- Arrive in Tel Aviv at 1:30PM
- Take shuttle to Jerusalem
- Spend night in Jerusalem
MONDAY and TUESDAY
- Sight-seeing in Jerusalem
WEDNESDAY
- Take bus to Dead Sea, then to Eilat, then to Jordan
THURSDAY
- Spend day in Petra
- Stay over in Wadi Musa
FRIDAY
- Wadi Rum (???), then head to Amman
SATURDAY
- Spend day in Amman
SUNDAY
- Fly back to the US
Are you now flying home from Amman?
Amman is a big city. Some people like it others feel it lacks some of the charm and unique characteristics of other cities in the region like Damascus or Cairo. Regardless, Wadi Rum is an entirely unique experience. If it were me and I was flying home from Amman I'd spend a day and a night in a Bedouin camp touring the desert. Eating a meal cooked under the sand and sleeping under the stariest night you've ever seen appeals to me a lot more than seeing yet another city. I'd skip Amman altogether.
Other than that it looks better.