Your analysis is too facile. Russia, unlike the US, has only Sunni enemies. And, by and large, radical Islam is a Sunni phenomenon — Al Qaeda, Hizb ut Tahrir (active throughout central Asia), the Muslim Brotherhoods, the Chechen and Ingushetian “nationalists” who bomb Moscow subways are all Sunni.
The sole example of Shi’ite radicalism is the Iranian regime, which is, alas, hostile to the US because the Shah had been one of ‘our bastards’ during the Cold War, and is run by a minority Shi’ite sect, the “Hasteners” or Ta’ajili, who think they can hasten the return of the 12th Imam by creating chaos. Traditional Shi’ites, whether ‘orthodox’ Twelvers like those in Iraq who objectively backed the US invasion or Ismaeli (or Sevener) Shi’ites by and large cause no one any trouble, and regard the Iranian regime with suspicion.
Russia’s backing of anti-Israeli groups is indirect and a function of support of the Alawite regime in Syria and their (I think mistaken) attempts to use Iran as a pawn in balance-of-power politics vis-a-vis the US. On the other hand in disputes over oil and gas resources in the eastern Mediterranean, Russia has been actively backing the Greek-Cypriot-Israeli consortium against Turkey, even sending warships to see off Turkish interference.
I do not think Russian policy in the Levant can be properly understood without considering the fact that Russia has traditional religious interests in the region. In pre-Soviet times Russia was the protector of Orthodox Christians in the region and pressured the Ottoman Sultans, sometimes sending warships, to ensure good treatment. Putin sees himself as restoring Russia to her traditional place, and as such will back the Israelis when they throw in with the Orthodox Greeks and Cypriots, but will also back the Syrian Alawites because they protect Christians from the enthusiasms of the Sunni majority.
Just so. Russia is returning to its historical role of protector for Christians (especially indigenous Orthodox Christians) in the middle east. The historical role of the United States meanwhile, is the dismantling of Christendom, to be succeeded by endless varieties of demented, rootless, secularized heresies.
There is also a factor in Russian thought that I think you missed. I think they actively play the roll of spoiler, taking any and every opportunity to make Western European powers and the U.S. weaker.