Alot of Jews in Europe are converts.. some were Romans that moved. For the most part though, most Middle Eastern Jews converted to Islam during the initial parts of the Arab conquests or to Christianity before that. Though most Christians in the region wound up converting to Islam in the region as well.
As far as moving back to Israel, it began at the turn of the century.. there was a massive Zionist movement with European Jews and with the Balfour declaration by the Brits in WWI it opened the doors to Jews moving to Palestine (which was set up as a nation under British rule at the end of WWI). That trickle continued, and during and after WWII lots of European Jews took off to America and Israel because they were being killed.
Once there they were under Palestinian rule they began making moves to set up their own state. They resulted to terrorists actions, killed alot of Palestinians, then tried to set up a two state system which the Palestinians rejected. The Jews then declared their own government got support from the US and Russia.. the Palestinians then took to arms in response and got their ass kicked. So Egypt, Syria, Lebbenon (well, sorta) and Transjordan then attacked Israel in response to this. They captured the Golan heights (Syria), Gaza (Egypt) and the West Bank of the Jordan River (Transjordan). Jordan then changed its name to Jordan (since they were on both sides of the river). However, the failed to crush the Israels so the Jews got their own state and the Palestinians fled to neighbooring states.
Later the soviets would feed Egypt and Syria bad intel that the Israelis were going to invade, so they built up their arms.. in response to this the Israeli's took the initiative and beat the living shit out of the Egyptians and Syrians and took back Gaza, Golan and the West bank. So now Jordan is stuck with the name Jordan even though their old name Transjordan makes sense again.
The Palestinian Liberation Organization set up shop in Jordan, eventually tried to kill King Husayn (or Jordan) with the help of Syria. Husayn asked the Americans for help, but the Americans couldnt get there in time, so the Israeli air force came to the rescue and saved Jordan... and they were the first state to recognize Isreals right to exist. This also got the PLO and Arafat kicked out of Jordan so they went to Lebbenon. The Israelis had none of that and invaded Lebanon sent the PLO to exile in Tunisa in the 80's. They would return to Israel as part of the Oslo accords in the early 90's.
From there Sadat would reinvade Israel to regain control of the Suez Canal. They couldnt afford the war with Israel so Carter steps in, negotiates peace. Isreal gives the Siani back to Egypt, Egypt recognizes Israel and that brings you up to the 80's.
