Could good relations have been maintained with both the Arab states and Iran?
How would Israel fare with Soviet support instead of Western support? Would the Soviets be willing to look the other way regarding an Israeli nuclear weapons program?
Depends what you mean by good relations. Iranian-Israeli relations are probably going to go on the same track as OTL (maybe the revolution is butterflied away, but maybe not).
Israel would probably get consistent Soviet support in the UN (the US was an ally of Israel, and still passed resolutions critical, while the USSR didn't do that for their allies). I think the Soviets would look the other way, but they also know that Israel wouldn't use nuclear weapons unless there was no other option. Israel would probably develop a nuclear program on its own without Soviet aid or anything.
I've actually been thinking about this one for a while, and one thing that I think would increase it's likelihood is if one of the major Arab countries, possibly Iraq or Syria, ends up going Communist. This would provide a Soviet-aligned Israel with a potential ally in the region, whilst serving to drive a wedge between the various Arab states and preventing them from uniting around Israel. This might also lead to a split in Ba'athism, between a left-wing pro-Communist faction that supports reconciliation with Soviet Israel and alignment with the Soviet Union, and a right-wing nationalistic faction that seeks to build ties with the West.
I also wonder how this is going to affect Israeli society and the Palestinians. If Israel gets really Communist and their position in the Middle East is reasonably secure, I can see Israel moving towards post-Zionism or bi-nationalism, what with national self-determination and anti-colonialism being fairly central tenets of Soviet ideology (at least in theory). The potentially quite large influx of Jewish migrants from the Soviet Union and other Warsaw Pact countries would help to placate the concerns of some of the more belligerently Zionist elements.
Yiddish might do better ITTL if there is a large influx of East European Jews who aren't necessarily as ideologically committed to Zionism and its fetishisation of Ivrit and restoring the old language.
Finally, assuming that Israel going Communist isn't enough to tip the Cold War in favour of the Soviet Union, what's going to happen to Israel if/when the Soviet Union falls.
Well, Iraq nearly fell a communist coup in I think 1959, so that might help you. Having a pro-Soviet or a communist Arab state might help, but I'm not sure. The Arab states were largely pro-American in the 1950s and 60s, and they still went over to the Soviets. The Soviets might make it so that if the Arab states want to get funding, they recognize Israel and make peace. If the Arab states go with the Soviets instead of the US, the Americans will be without allies in the region and they'll be desperate for an ally/asset in the region, and try with Israel.
Israel isn't going to go Communist, post-Zionist or bi-nationalist. They'll get Soviet support, but probably remain socialist (like India or maybe something like Yugoslavia). They didn't in OTL, and there's no reason to think they will in TTL. They'll be under the Soviet umbrella, but keep their distance because there's no support for a Communist Israel in Israel (the Communist parties in Israel never won more than a few seats). Plus, if Israel is under Soviet influence, the Soviets will make Zionism a national liberation movement (plus they get to say they're the "protectors of the Jewish people, who have been so unjustly and brutally persecuted by the bigoted and capitalist lackeys of world imperialism"). With Israel under the Soviet shield, there will be pressure to allow Soviet Jews to come to Israel; and with a very good Israeli defense industry, they might tie Soviet Jewish immigration to Israel with either Israeli defense industry secrets, or with Western military secrets. Hell, the Soviets would maybe push for a naval base in Haifa, to counter the U.S. Sixth Fleet directly, and to get a warm-water port.
Yiddish is all but dead. Soviet Jews spoke Russian, and very few actually spoke Yiddish. Yiddish isn't going to make a revival, it's going to remain a minor language in the USSR, US, and Israel. And Soviet Jews were actually quite Zionist (not all, but a good majority).
When/if the USSR collapses? Israel migrates over to the US (which will no doubt have been trying to woo them over to the West for decades if they lose Arab states to Moscow) like India did. They'll probably get some concessions because the US would be desperate for an asset in the region, like weapons deliveries or unconditional support in the UN.