The immigration stamp is placed on a separate sheet of paper. There's no stamp in your passport. You can easily travel to Iran the other day.
Arrive late at Ben Gurion airportNo worries, even if you arrive at 4 a.m. there are lots of sheruts (shared taxis) around which operate regularly all around the clock. It takes half an hour to get to Tel Aviv and approximately an hour to Jerusalem. Be aware; it might take a while to drop every passenger off at their place.
ClimateAvoid travelling during summer months. It is just too hot from June to September, especially in the Jordan valley and Dead Sea.
FoodFalafel, Shawarma and the Israeli breakfast were our favourite dishes.
TravellingRenting a car is cheap. Israel is a small country. You get nearly everywhere within a few hours. Driving in Nazareth is a nightmare. If you don't use the navigator app, you easily get lost and if you use it, it guides you through narrow and steep alleys.
SafetySounds odd but I felt safe in Israel aswell as in Palestine. You easily walk around at night: Hardly any beggars, neither drunks nor drug addicts. Citizens are easy and relaxed. I didn't experience any tension between the different religious communities. ....Just my impression.
PalestineYou might not be allowed to enter Palestine with a hire car. Double check with the rental car company. Fix a price when taking a guide (to drive you around) and make sure he understands you and you do understand him.