
Right wing and Revisionist Zionist. However supporting a two-state solution, but by exchanging majority-Arab parts of Israel for majority-Jewish parts of Palestine, without forcibly resettling peoples. Staunchly opposed to the religious politics and privileges of the Orthodox Jews (particularly their exemption from military service). Supportive of mandatory military service for all Israelis, whether Arab or Jew. Yet also says any arabs who wish to be citizens must swear an oath of loyalty.
Yisrael Beiteinu is thus interesting as it is a right wing war hawkish party which nonetheless supports a two-state solution, unlike Likud or the other more Mizrahi-supported parties. I think this is in a large part due to it reflecting the secular practical desires of the Soviet immigrants, rather than the more religious conservative desires of the Mizrahim and ultra-orthodox. They don’t care as much about the idea of a religious promised land.
It heavily depends on the country. I mean the general rule is that you don’t get hundreds of thousands of people leaving their entire livelihoods behind to live in squalid refugee camps in the Israeli desert willingly. In some cases it was an official government expulsion (like Egypt). In some cases a government expelled the Jews as part of an explicit agreement with the Israeli government (Iraq). In some cases Israel just airlifted the whole community (Yemen).
In most cases the exodus was a mixture of the pull of a genuine desire to go to Israel, and the push factors of increasing community violence from Muslim neighbors. Morocco and Tunisia were two governments who actively tried to retain their Jewish communities (and still have some today), but community violence continued to push them out.
Jewish emigration from Iran has never been allowed by its government (many Muslim governments banned it early on). However most of the community still left the country, and the Israeli government had programs to smuggle Jews out of Iran to Israel (they had similar stuff for Morocco before emigration was allowed).