Why would it want to to be a member of NATO? Leaving aside the issues of the former Eastern Bloc countries being very uncomfortable about Russia, would Russia really give up 500 years of xenophobia and be happy to be a sub-ordinate part of somebody else's alliance?
Because 500 years of xenophobia didn't really work out that great for it?
Also if your part of a military alliance, that alliance should be less threatening to you.
And protect you from other threats.
Also, sub-ordinate? Not really aware of the US really domineering, say France.
As for the EU - why does the EU want to support even more poor Eastern European farmers / factory workers. It's not like Russia is going to provide raw materials at below market cost.
Why would NATO want to get involved in Russia's border wars?
Might provide better investment oppertunies?
And as for Russia's border wars, a more secure Russia might be a better neighbor.
And then you have the very concept of collective security. Bigger is stronger and more likely to not have to fight, and if fight, win.