As announced, I went to Israel and Palestine in June. Here are some tips.
I rented a car with a major rental company (Sixt, which is called Shlomo there) in Ben Gourion airport.
As said before, you cannot take a rental car into areas A (full control of Palestinian Authority) and some areas B (shared control). The main reason is insurance. In case of accident, insurances will not cover you if you happened to be in area A and some roads in areas B.
Saying that, I saw many rental cars in Bethleem and Jericho (area A), and there was no one to check. Being a bit overcautious, I did not drive my rental car into those areas.
Roads that you cannot take with you rental car are the same Israeli citizen are not allowed to take, and they are clearly marked with huge red signs saying "This road leads To Area "A"... The entrance for israeli citizen is forbidden, dangerous to your lives and against israeli law".
You can plan your trip on google maps, as the "forbidden" roads are ommited.
As a foreigner, you have the right to go beyond, and according to people I met in June, and perfectly safe to travel (except for Gaza Strip of course).
Most sites in the south are reached from Bethlehem. To get to Betlehem, take an arab bus 21 from Damascus gate in Jerusalem, leaving every 20 minutes or so. It costs 8 NIS. It goes to Beit Jala neighborhood of Bethlehem, where it is a 15 walk to Church of Nativity. No check-point to pass on the way in. Back to Jerusalem, you pass a check-point but as a foreigner, most of the time, you just stay in the bus and wave you passport at the soldiers.
About the TL sites:
- Ancient Jericho: Tell es-Sultan: there are numerous guided tour from Jerusalem, or you can take a shared taxi. I went by car to Almog junction, on the road number 1 that goes to the Dead Sea. There is a restaurant and a petrol station where you can park a car, and from there, hail a shared taxi (3 NIS) or a regular taxi (price to negociate, probably 20-40 NIS) to central Jericho. From there, a shared taxi to Tell-es-Sultan for 2 NIS.
- El Bariyah: wilderness and monasteries: This project covers a widespread area which crosses all three A, B and C areas.
Herodion, Saint-George Monastery and Nabi Musa are in area C and can be reached with your rental car (Saint-George is a 30 walk down a wadi).
Mar Saba Monastery is in area C, but the road that leads to it is in area A, so you need to take a taxi from Bethleem (taxis are waiting everywhere in the Bethleem area, and a half-a-day trip from 100 NIS, to negociate; the driver will propose to stop at Herodion on the way; you can also stop at Deit Theodosios, but I was refused entrance because I am not Orthodox !)
-Jerusalem Southern Terraced Landscape (Battir): It is in area B and C, but on a "red road", so you cannot drive your car there. Easiest is to take a cab from Bethleem (around 50 NIS roundtrip) or take a shared taxi from "cinema Square" in central Bethlehem. Ask for Battir village. Ancient roman fountains are within the village, and olive terrace are all around the village (a road crosses the village and leads into a nice valley for a few hundreds meters, just ask the taxi driver to get you there, or walk 15 minutes).
- Old town of Hebron: I went there after my visit to Mar Saba with my taxi driver (for an additional 200 NIS). He showed me the old town and the divided city. Passing check-point as a foreigner is rather straightforward. The highlight however is the Cave of Patriarchs, divided into two parts, muslim and jewish, with bullet-proofed windows in-between. As a foreigner, you are allowed to visit both sides, passing numerous metal detectors.
- Qumran is in area A, and is managed by the Israeli National Parks. It is easy to reach from the road to the Dead Sea.
- I was planning a guided visit to Nablus, Sebastia and Mount Gerizim, but it was cancelled at the last minute for lack of participants, so I cannot comment on those two sites.
- Umm Al Rihan forest is in Area C, and seems to be easy to reach, but I lacked time to get there.
- Finally, the three remaining TL sites are in the Gaza Strip and not recommended for travellers at the moment.