No, they do not. Culturally Russia has always ping ponged back and forth between seeing itself as European and not European. Currently Russia is on the "we're distinct from Europe and have our own unique and proud history" side of that equation. If you want more on this (extremely complicated) subject, read into the various Tsars that attempted to "modernize" and Europeanize Russia, such as Peter the Great. The backlash after his death towards his European preferences says a lot.
Religiously Russia is an Orthodox country, and it split off from Europe's Catholicism nearly a 1000 years ago. This split was both religious and political, aligning roughly (very roughly) with the divisions of the Roman empire into East and West. The Eastern (Orthodox) branch was originally centered in Constantinople (modern Istanbul), which even today we'd identify as a distinctly Eastern city. That alone should tell you why Russia sees itself as distinct from the rest of Europe.
Meanwhile the rest of Europe is either Catholic or Protestant, which itself split off from Catholicism half a millennium after Orthodoxy did. They're just not that close religiously.
> Seems like Russia and all countries that share similar cultures would be allies against China, with her 1.5 billion people . . . ???
Because international diplomacy and conflict isn't really as simple as pointing at borders and population counts. Yes, there has been some conflict between Russia and China, but by and large their population centers are a long, long way away from each other. There's a reason why Russia has fought basically three wars with China and dozens with Sweden alone; geography.
Meanwhile there are historical and personal reasons for Russia to turn West and specifically to Ukraine. Russia has a very, very long history in attempting to seize and hold Ukraine, as it's alternated back and forth between being a Russian owned territory and not. Take a look at the Kievan Rus, the ancestor state to both Russia and Ukraine, and coincidentally where Russia converted to Orthodox Christianity.