Hello, Ines.
I will try to answer a few of your Israel questions.

Just taking a taxi from Eilat to the Dead Sea will cost hundreds of dollars, if you could find someone who wanted to do it! Stopping at the Dead Sea and then continuing to Jerusalem with a different taxi would also be very expensive.
A tour guide with his/her own car would also cost hundreds of dollars for the whole day. Yes this can be done by bus for much less money but you need to check bus schedules in advance.

There is regular public transportation starting early Saturday evening from Jerusalem to Tel AViv, but are you sure you don't want more time in Jerusalem on Sunday when everything is open the whole day?

The Old City of Jerusalem is open for business as usual on Saturday, except for the sites in the Jewish Quarter. There ate many places in the New City that are open as well, such as the Israel Museum, Museum of Islamic Art, etc. (you have to check their web sites for Saturday hours).

Visiting the Dead Sea on the way from Eilat is difficult and costly. A good idea is to take the Abraham Hostel "tour" from Jerusalem that goes to both the Ein Gedi Nature Reserve and Massada:
http://www.abraham-hostel-jerusalem.com/en-us/activities/tours.aspx
This "tour" provides only transportation and there is no guide, but you can get audiophones at Massada.

Many thanks for your replies.
I have decided to take the bus from Eilat to Jerusalem, and checked the bus time table (7, 10, 2 and 5 pm) I wanted to stop at the dead sea and then continue but I am realising now that is not possible or too expensive..
Do I need to book in advance this bus to Jerusalem or can I buy the tickey before the bus departs?
I will stay from Monday to Saturday in Jerusalem and then go on Saturday afternoon to Tel aviv, I am flying backon Sunday evening to the UK.

Do you know how long will take from Petra to Eilat including crossing the border and how much can I expect to pay?

The Jett bus from Amman to Petra, do I also need to book it in advnce?

Many thanks,
Ines

Amman to Petra: Post your question on the Jordan forum.

Eilat to Jerusalem: you should book in advance by phone. Here is the link:
http://www.egged.co.il/Eng/main.asp?lngCategoryID=2771

Jerusalem to Tel-Aviv on Saturday: there is a shared taxi service (called "sherut") that runs all day; there is no bus service.

You may prefer to go to Tel-Aviv on Friday because Jerusalem is rather quiet on Friday evenings, while Tel-Aviv is the opposite. You can take either the #405 bus to the Tel-Aviv CBS or the #480 to the Arlozorov terminal in north Tel-Aviv (I recommend the #480, unless you are going to Jaffa).

You have to be at the airport 3 HOURS before your departure. From Tel-Aviv you can take either a taxi or the train (there is no sherut). Train schedules:
http://www.rail.co.il/EN/Pages/HomePage.aspx
Note that the airport is located between Jerusalem and Tel-Aviv.

I stayed at the Dead Sea Marriott in Oct 2010 and just loved it.

I didn't find Amman very interesting, and don't know what you'd find to do for just a couple of hours.

Staying in a camp is an activity associated with Wadi Rum, not Petra. Do you have a guidebook yet? Try Lonely Planet or Rough Guide.

Same here thursdaysd.

Amman is definitely not interesting but Jerash is quite impressive and worth a visit if you have the time.

Hi It's about 11/2 to 2hrs from Petra to Eilat by taxi. We negotiated a fare of 130 JD for the ride. Crossing the border is easy. You will probably already have a visa to cross. It was simple. everyone speaks English at least somewhat.