It appears that it's safe to rent a vehicle and do a self drive through Israel. What about crossing over into Jordan? Is it recommended to have hotel reservations the beginning of August? Thank you.
Actually, the Israelis are harsh drivers, but once you get the spirit - you might even like it.
The roads are quite good. Loads of traffic, however, especially during rush hours (7-9 AM, 4-7 PM).
Renting a car will solve you the headache of getting from point A to point B, as most of the out-of-town tourist spots lack normal public transportation (yet boast plenty of parking space). Israelis love their cars and hate their busses, and they have all the reasons for it.
The only places in Israel a car would be a burden at are Tel-Aviv and Jerusalem city centers. There you can safely park at the hotel and go walking (or take a bus, or a cab). For all other places - DO take a car.
You won't be able to drive it to any neighboring country, neither will you be allowed to go to the Palestinian Authority. Cities like Ramallah, Gaza, Beth-Lehem etc. are off-limits for any car with an Israeli licence plate. Israeli settlements in the occupied territories are safe to visit, however, yet it's boring there and nothing special to do.
