French defense queen side castle

@1
"is there any French defense variation, where black can castle without limitation on queen side?"
++ Yes, there are some. Here is an example:
https://www.chessgames.com/perl/chessgame?gid=2164701 

If you are able to blockade the pawns on the queenside, then it will be very hard for White to a breakthrough on the queenside. I'll show a line to demonstrate my point

Yes, but you cannot "premeditate" that you are going to castle Queenside.  You need to do what is appropriate for the situation.

For example:  After 1.e4 e6 2.d4 d5:

Exchange - After 3.exd5 exd5, Black almost NEVER castles Queenside.  Due to the open 3-file, Black needs to Castle ASAP, and with only 2 pieces to get out on the Kingside, Kingside Castling is pretty much automatic, like 4.Nf3 Nf6 5.Bd3 Bd6 6.O-O O-O

Advance - After 1.e4 e6 2.d4 d5 3.e5 c5 4.c3 Qb6 (If you are going to play the 5...Qb6 line, play it on move 4 to avoid the 5.Be3 line - Here 5.Be3? drops the b-pawn) 5.Nf3 Nc6 and now, it all depends on White:

A) After 6.Bd3, Black NEVER castles Queenside.  6...cxd4 7.cxd4 Bd7 8.O-O Nxd4 and now after 9.Nxd4, 9.Nbd2, or 9.Ng5, Black will always castle Kingside or not castle at all and go to f8 with the King

B) After 6.Be2, Black almost always castles Kingside

C) After 6.a3, it depends on what Black does to follow:

C1) After 6...Nh6, Black almost never castles

C2) After 6...c4, Black almost always castles Queenside, but you really need to know the theory here.

Tarrasch - Black basically NEVER castles Queenside here.  Almost always it's Kingside Castling.  The lone exception is in the 8...g5 line of the Universal system where Black usually doesn't castle.  That is 1.e4 e6 2.d4 d5 3.Nd2 c5 4.Ngf3 Nf6 5.e5 Nfd7 6.c3 Nc6 7.Bd3 Be7 8.O-O g5.  Otherwise, in the Open Tarrasch (4.exd5), Closed Tarrasch (3...Nf6), or the Guimard (3...Nc6), Black castles Kingside.

3.Nc3 - It all depends on the lines played by both White and Black:

A) After 3...dxe4, Black almost never castles Queenside.  It is Kingside or not at all.

B) After 3...Nf6, you have the following:

B1) In the Steinitz (4.e5), it is basically Kingside for Black always, and for White, he can castle either way, though Queenside is more popular

B2) After 4.Bg5, you have:

B21) In the Classical, 4...Be7, Black always castles Kingside

B22) In the MacCutcheon, 4...Bb4, Black either castles Queenside or not at all - solely depends on the line played by White.

C) After 3...Bb4, again, it all depends:

C1) Against moves other than 4.e5, it is always Kingside or Center for the Black King

C2) After 4.e5 c5, if White plays something other than 5.a3, the King almost always castles Kingside.  If 5.a3, then after 5...Bxc3+ 6.bxc3, Black has:

C21) 6...Qa5 7.Bd2 Qa4 - Here the Black King either goes Queenside or winds up on f8

C22) 6...Ne7 (The main line), whether Black goes Kingside or Queenside depends on the decisions of both sides:

C221) After positional lines like 7.a4 or 7.Nf3, it all depends on what Black does with the c-pawn.  If it advances to c4, Black usually castles Queenside.  If it trades on d4, Black usually castles Kingside

C222) After aggressive lines like 7.h4 or 7.Qg4, Black can wind up anywhere.  After 7.Qg4, 7...O-O is a line, 7...Kf8 is a line, and in the case of 7...Qc7, Black could wind up in the center or on the Queenside.

So as you can see, YOU CAN NOT PREMEDITATE where your King is going in the French Defense!

@ThrillerFan Thank you very much for your answer. 

I have tried castling in advance variation on queenside after 4.c5 after blocking my queenside with pawn pushes. Usually it ended horribly because my opponents sacks piece to break my pawn chain and win in endgame. Just like in game mentioned by tygxc.

I have never imagined that you can castle queenside with Nc3 variantions.