Observe this position. 

7.bxc3 is better. That's all I got though...

This isn't bad for white.  There will be a pawn capture at some point between c4 and d5, eliminating the doubled pawns.  If you don't want a weak pawn on c3, put the queen on c2.  

Black certainly doesn't have an attack, and arguably he's behind in development.

White should react to 5 ...Ne4 with either just 6 Qc2 or 6 Qa4+ Nc6 7 Qc2, with a normal advantage.  Black is far better off playing for a more mainstream NID position with 5 ...c5 or ...0-0.  It gets hard for White to prove any advantage in most of those lines.

I'll roughly agree with the earlier comments. Also note this is not a Nimzo yet. 4.Nf3 Nf6 5.Bg5 and I think this is the Ragozin? Which is fine for white. The doubled pawns are again no problem, if Bxc3 then cxd would undouble at any time.

Notice that you don't _really_ have doubled pawns, because you can exchange one of them at any time with cxd5, and if they have played c6 to back up the centre you then have the option of playing c4 again! I enjoy positions with doubled c-pawns although I normally play the Qc2 Nimzoindian variation.

Move 4. Qc2 is completely the wrong move, if attempting to open QG. It is of course Nf3 prior to any other move. There is no self persecution if B decides to take, as the doubled pawn can be released later. Knights out is the intent, and attacking K'side castle with lines is always the way. Qc2 is a lost move in maintaining position as attacking white plays. -J.

White has a strong center , better bishop and a half open b-file.If white is a decent endgame player, this position can be really enjoyable for him. The only plan that I can see for black is c5 b6 Bb7, but even then white will have too much pressure on the queenside with a4 a5.