Subtle rook move in the French Steinitz variation Ra7?!

After 1. e4 e6 2. d4 d5 3. Nc3 the most stable answer for me is still 3. ... Nf6. This is a matter of taste, playing 3. ... Bb4 here is of course possible, however this results in a sharp game and white players seem to be able to come up with nasty tricks in this line for which I'm not prepared.

When you play 3. ... Nf6 then 4. e5 is almost certain, you will have a depressed position and white will attack on the f-line against the pawn on e6. I always had problems with what to do with the "wrong" light square bishop here. If after the natural a6, b5 expansion I develop it to b7 hoping that the diagonal will open then it is missing from the defense of the e6 pawn. If I develop it to d7, then it fails to protect a6 once the b5 pawn is gone. At the same time I definitely want a rook on the c-file so I have some counter attack. How can I meet all these requirements? I came up with the move Ra7?!. This allows the rook to come to c7 without moving the ls bishop and also provides some lateral defense to the 7th rank should white attack on the king side. The engine is not very happy about this move, however the experience of many games taught me that this is a good practical choice.

In the example game I think white was a bit slow with the king side attack, yet the rook finally proved itself allowing a fatal blow while the bishop secured both e6 and a6.

en.lichess.org/2lkxM0rH/black#20

Why did white take out of the centre 11 dxc5?
Why did white castle queen's side 13 o-o-o instead of King's side 13 o-o?
What good does 14 h4 do?

@sakkozik gg wp. That ra7 move had me somewhat confused and intimidated, reaffirming the thought that I don't understand this opening at all..
@tpr
I'm new to this opening, used to play the advance variation. So I'm pretty clueless, and spent way too much time in the opening here. To answer your questions, I castled queenside because that's the only plan I'm familiar with in this opening. White usually follows up with h4-h5 afaik, but not sure how great that is when black hasn't castled yet. I just didn't know what else to do..

The reason I captured on c5 was because I wanted to castle queenside, but if I do that right away black can play c4 which I think is a problem.

Interesting to get feedback from black as well as white.
Normally king's side castling o-o offers the advantage that the King stands safer and that the rook on f1 helps to push f5.
h4-h5 is only useful after black plays o-o or Kf8. Normally the push f4-f5 is better.
Capturing on c5 and then castling queen's side is far more dangerous for white than castling queen's side after black plays ...c4. The open c-file is much more dangerous than the pawn advance ...b5-b4.
This game illustrates the dangers for white of dxc5, of o-o-o and the harmlessness of h4 if black has not yet castled ...o-o.
9 a3 seems incompatible with the plan o-o-o, as it weakens the King. There were many alternative moves 9 Be2, 9 Ne2, 9 h4, 9 Qf2, 9 Bd3, 9 dxc5.

@sakkozik Usually when black is clearly preparing an offensive on the queenside, then just instinctively want to take away the knight from the square c3, and make the pawn move c2-c3.