[Goose]

The main connection is Iran. 

Iran funds and supports Hamas for ideological reasons, partly real and partly faked to justify the regime’s continued power. In addition, supporting Hamas allows Iran to project power in the region while maintaining non-plausible deniability that some believe despite the thinness of the veneer.

To the best of my understanding, Iran supports Russia against Ukraine for a different reason, which is that Russia can actually provide material support to Iran, as well as veto UNSC resolutions on Iran’s behalf. This is more Machiavellian than ideological.

I think you are correct about the above. Chaos in the middle east is viewed in Iran as a "good thing". Iran has delusions of becoming the dominant regional power. Preventing other powers from concentrating on Iran as the "enemy" is good in Iranian eyes. As long as they aren't "frozen out" by peace breaking out in the middle east, Iranian ambitions have at least some local credibility.

I don’t think Russia cares about the Israeli/Palestinian conflict (though I could be proven wrong).

Russia cares very much about the Israeli/Palestinian conflict because the Russians hope to return to the days when their naval presence in the Mediterranean Sea was a significant problem for NATO. That is why Russia is in Syria. They had been forced by circumstances to pull all their forces out of the Mediterranean. That allowed the US to gut the 6th fleet and move essentially all of the (reduced) assets to the 5th fleet in the Persian Gulf. 

Russia's return to the anchorages in Syria in late 2015 (Moskva and two escorts plus an unknown number of Kilos periodically) really shook up NATO South. NATO South is now basically the Italian Navy, and this responsibility weighs heavily on them. They took it over without a fuss and did the job the 6th fleet used to do. Granted, Russia isn't what they were, but Italy wasn't really resourced to to do this task. They get no credit in public, but in private the US Navy knows the score. That is one reason that the Ford was "exercising" with the Giuseppe Garibaldi. We need those guys.
From https://en.wikipedia.org/wiki/Russian_cr...yag_(1983)

In early January 2016, after a visit to India, Varyag entered the Mediterranean Sea through the Suez Canal to be deployed off Syria′s shore replacing sister ship Moskva,[23] in support of Russia's air operation in Syria that had begun in autumn 2015. The ship was named flagship of the Russian naval task force positioned in the eastern Mediterranean.[24]

She didn't stay all that long, but it is worthy of note that she (unlike Moskva) was "fully modernized" and fresh out of the yards at that point. This replacement created another tsunami of activity in NATO South.

Russia believes that it's presence in the middle east and Mediterranean creates big problems for NATO. Anything that weakens US ally Israel makes it more difficult for the US to counter this Russian influence. It also makes it more difficult for other middle eastern states to oppose Russian ambitions in the area. Lebanon is a poster child for this effect, as is Yemen. As long as Hamas has power in Gaza, it can stir the pot in Israel and elsewhere in the middle east. Russia likes that a lot. This strategy goes back to the days of Nasser/UAR days which is what actually started the US/Israel alliance. Despite a few "ups" and many "downs" for this approach, Russia still believes conflict in the middle east is to their advantage. In the past, they also had financial incentives they could provide to their client states. They basically can no longer do this, but they do have some military support that is an attraction. Iran is operating somewhat as a Russian "proxy" in the middle east, but for their own reasons, not Russian ones. Russia is getting the benefit at a very low cost to itself.

[Goose]

The main connection is Iran. 

Iran funds and supports Hamas for ideological reasons, partly real and partly faked to justify the regime’s continued power. In addition, supporting Hamas allows Iran to project power in the region while maintaining non-plausible deniability that some believe despite the thinness of the veneer.

To the best of my understanding, Iran supports Russia against Ukraine for a different reason, which is that Russia can actually provide material support to Iran, as well as veto UNSC resolutions on Iran’s behalf. This is more Machiavellian than ideological.

I think you are correct about the above. Chaos in the middle east is viewed in Iran as a "good thing". Iran has delusions of becoming the dominant regional power. Preventing other powers from concentrating on Iran as the "enemy" is good in Iranian eyes. As long as they aren't "frozen out" by peace breaking out in the middle east, Iranian ambitions have at least some local credibility.

I don’t think Russia cares about the Israeli/Palestinian conflict (though I could be proven wrong).

Russia cares very much about the Israeli/Palestinian conflict because the Russians hope to return to the days when their naval presence in the Mediterranean Sea was a significant problem for NATO. That is why Russia is in Syria. They had been forced by circumstances to pull all their forces out of the Mediterranean. That allowed the US to gut the 6th fleet and move essentially all of the (reduced) assets to the 5th fleet in the Persian Gulf. 

Russia's return to the anchorages in Syria in late 2015 (Moskva and two escorts plus an unknown number of Kilos periodically) really shook up NATO South. NATO South is now basically the Italian Navy, and this responsibility weighs heavily on them. They took it over without a fuss and did the job the 6th fleet used to do. Granted, Russia isn't what they were, but Italy wasn't really resourced to to do this task. They get no credit in public, but in private the US Navy knows the score. That is one reason that the Ford was "exercising" with the Giuseppe Garibaldi. We need those guys.
From https://en.wikipedia.org/wiki/Russian_cr...yag_(1983)

In early January 2016, after a visit to India, Varyag entered the Mediterranean Sea through the Suez Canal to be deployed off Syria′s shore replacing sister ship Moskva,[23] in support of Russia's air operation in Syria that had begun in autumn 2015. The ship was named flagship of the Russian naval task force positioned in the eastern Mediterranean.[24]

She didn't stay all that long, but it is worthy of note that she (unlike Moskva) was "fully modernized" and fresh out of the yards at that point. This replacement created another tsunami of activity in NATO South.

Russia believes that it's presence in the middle east and Mediterranean creates big problems for NATO. Anything that weakens US ally Israel makes it more difficult for the US to counter this Russian influence. It also makes it more difficult for other middle eastern states to oppose Russian ambitions in the area. Lebanon is a poster child for this effect, as is Yemen. As long as Hamas has power in Gaza, it can stir the pot in Israel and elsewhere in the middle east. Russia likes that a lot. This strategy goes back to the days of Nasser/UAR days which is what actually started the US/Israel alliance. Despite a few "ups" and many "downs" for this approach, Russia still believes conflict in the middle east is to their advantage. In the past, they also had financial incentives they could provide to their client states. They basically can no longer do this, but they do have some military support that is an attraction. Iran is operating somewhat as a Russian "proxy" in the middle east, but for their own reasons, not Russian ones. Russia is getting the benefit at a very low cost to itself.