Getting Israel to join the Arab League is impossible. Period. As long as Israel is a Jewish state. The only way I can see this as possible is Israel losing the War of Independence, but then the Palestinian state in its place would include the West Bank and Gaza.
As for the other stuff...If Somalia wins the Ogaden War with Ethiopia 1977-78, then you could see the African borders. Most African states are unstable anyway, so it shouldn't be hard for some kind of civil war to take place in the countries with Arab populations and the Arab regions secede and are promptly recognized by the Arab League. And Iraq winning the Iran-Iraq war might result in the border with Iran. Maybe a tighter Arab League beats the Israelis in the War of Independence, then provides military aid to the Iraqis and Somalis when they get into fights. I think you'd need a tighter Arab League structure to make this map possible. A collective security clause, like Article 5 in the NATO Charter, would be very helpful.
EDIT: Actually, this map isn't far from the OTL Arab League borders. The only differences I can see are the inclusion of Chad, Western Sahara, and Israel. And some border regions. Granted, Israel is a pretty damn big exception, but still.