Personally I don't think white has much in these lines and I note that Topalov and Ivanchuk have played this.

11 bf4 bb4 12 c3 be7 seems fine for black - I wouldn't be shocked if Kramnik has this as backup in case his Petroff suffers.

Good question !!

I always thought that after 12.0-0-0 black must play 12..Bd6 and after 13.Ne4 till now, I played the move 13..Bc7 waiting for what white is going for (Kb1; c4; Nxf6.....)

Till this post I never had 13..Be7 in mind, but I must agree surely a playable move in this position.

I've often noticed that in the 3. Nc3 variation, Black plays Be7 instead of the seemingly more active Bd6. My first thought was that it was directed against Ne4, but after a simple exchange of knights and Nf6 it seems fine. My next thought was that Black is anticipating d5 after c4 and wants to put more pressure with a rook but this seems kind of unclear. Next, I considered that it was directed against g4-g5 but again this is much more in the future and more active play might compensate. Basically, I have a lot of ideas but I still don't know why Be7 is preferred so overwhelmingly in this position:

1. e4 c6 2. d4 d5 3. Nc3 dxe4 4. Nxe4 Bf5 5. Ng3 Bg6 6. h4 h6 7. Nf3 Nd7 8. h5 Bh7 9. Bd3 Bxd3 10. Qxd3 e6 11. Bd2 Ngf6 12. O-O-O Be7