The answer to your original question will sound a bit complicated.
You cannot take an Israeli rental car into areas that are under the complete control of the Palestinian Authority - areas that are called Area A. The includes Bethlehem, Jericho and most of Hebron.
Hebron becomes a bit complicated. If you are only going there to see the Cave of the Patriarchs and you keep the car on the Israeli side, then you can drive your rental car there. You cannot take it across the checkpoint to the Palestinian side.
That being said - you will probably be much more comfortable taking the bus. The route is a bit complex with some "interesting" twists and turns and places that you MUST go a certain way and not the straight way. It's one of the few places that I would not recommend you drive on your own.