You are correct saying that Russia doesn't belong to the West and there are certain reasons for that but to me the whole thing is not so much about the culture based on religion factor. Russian elites may have contributed to the development of 'Western thought' (literature, music, science) but clearly not 'Western society'. Russia over centuries clearly developed it's own authocratic model of ruling (in which far-East bonds are most clearly visible). Like in physics heavier bodies attract other, smaller ones. And Russia clearly was large enough to feel entitled to go her own, separate way. Every time I have contact with Russians I somehow feel stubborn pride and conscious choice that has been made about it. Putin also very well epitomizes these features of Russian psyche and the dream of baroque greatness to majority of Russians. Maybe that's also where religious and political perspectives meet...? I don't know.
Check Charles Murray's "Human accomplishment" book. In that book he has maps - about cultural contributions, where they were coming from. Suffice it to say, from russia these were pretty small.
This is cracked. Russia looked to Louis XIV's France for its autocratic model. French, not Mongolian, was the language of the Russian royal court.