Israel no longer stamps any passports. You get a little card on entry that you have to keep with your passport until you leave. Anyway, there is no problem going to/from Israel and Turkey or Israel and Jordan.
The big fall holidays beginning with Rosh Hashana start on September 13 and end on October 5 this year. This is high high season in Israel, everything is crowded since most Israelis are also on vacation for at least part of that time. There is limited public transportation on the eves and holidays. Kids go back to school on September 1. Depending on how long you want to stay in Israel, August 30-September 10 would be good. Before that it's the end of summer holidays and many Israeli families are on vacation the last two weeks of August along with all the tourists. After that, its holiday time.
Where you stay in Israel depends on how much time you have and what you want to see. You can easily travel (Sunday - Thursday) by train between Tel Aviv and Haifa and Acre (Akko) which I recommend seeing. To visit the areas around the Sea of Galilee and Golan Heights (lots of great stuff up north), you need to rent a car or take big-bus tours. The area is not at all well served by public transportation. You can visit the Dead Sea on a day trip from Jerusalem - by car, with a tour, or using the public buses. So you could see a lot from a base in Tel Aviv and another in Jerusalem.
You can cross the border from Israel into Jordan by land, but it may be faster and easier to fly. The border crossing nearest Jerusalem is not primarily for tourists and it can be a lengthy process.
Turkish Air has 5-6 flights a day from Tel Aviv to Istanbul. For flights to Jordan, try Royal Jordanian. Both must have flights between Amman and Istanbul.