Black's system focuses on main strategical idea: to exchange light square bishops.
Why ? ... well, another idea in Winawer is to hinder bishop pair activity (because of the choice of Winawer: to trade an active bishop pair for a pawn structure).
Pawn play (Kmoch-like explanation style): usually this can be done by reaching some Benoni closed center with f5 or c5. After f5, then White has choice to open kingside wing early with exf6 or do it later with a g4 pawn push. The c5 pawn push is good too to attack center pawn chain or to close all center with a c4 push.
Dream of exchanging light square bishops...can it be real ? Yes, but it takes time..and that's why White usually plays on the kingside, to win something there. For example: if we think about Petrosian's line: 4..Qd7 5.a3 Bxc3+ (Petrosian idea was Bf8...and you have not the Winawer trade as given before: bishop pair vs pawn structure) 6.bxc3 b6 (6...Ne7 7.Qg4 Ng6 - very bad idea is to play Winawer with tempi less: 7..oo 8.Bd3 7.Qg4 f5 8.exf6 Rxf6 9.Bg5 - 8.Bd3 and attack for White is sooo easy, playing normal moves for a wing attack if 8...oo and if 8...c5 9.h4 c4 10.h5 Ne7 11.Be2 +-) 7.Qg4 f5 8.Qg3 Ba6 9.Bxa6 Nxa6 10.Nh3 Nb8 11.Nf4 +/- now knight has reached one excellent square and pressure is on weaknesses e6 and g7 pawns.
Comeback to system after 4...Ne7 5.a3 Bxc3+ 6.bxc3.
Most aggressive is now: 7.Qg4 where Black has no time, and need to play 7..Ng6 (7...Nf5 8.Bd3 and too late for Ba6 now because of Bxf5 ; 7...oo? 8.Bg5 Qd7 9.Bd3 ; 7... Kf8 (Euwe's idea as given by Suetin) 8.Nf3 Ba6 9.Bd3!? Bxd3 10.cxd3 c5 11.dxc5 bxc5 12.oo and now 12...Nbc6 13.Rb1 and 12...Nd7 13.Bg5 (Khalifman))
Line to follow is now 8.h4 with focus on hindering Black not to long castle. One good book about it, is Shirov's Fire on Board 2.
8...h5 9.Qd1 (9.Qg3 Ba6 10.Bxa6 Nxa6 11.Qd3 Nb8 12.Bg5 Qd7 is the common position where Black had no time to play Qa4 and long castle....and now 13.Ne2 Nc6! is right plan for Black: blockade with pieces on light color squares Petrushin-Vaisser,Volgodonsk 1983 ... if 13...c5 then White has plan with 14.f4 and an attack on f5 square with help of Ng3 and oo) Qd7 10.Bg5 (better than 10.Bd3 or Be2 where Black has Ba6) Ba6 11.Bxa6 Nxa6 12.a4 (Dom) (12.Qd3 Qa4 and now Black has his counterplay on c file with a later c5-Rc8 and sometimes Nc7 to cover e6 after f4-f5-exf5 moves ; 12.Ne2 Qa4 13.Rh3 c5 or 13...Qc4 Balcerak-Lalic, 2006 ) c5 13.Qd3 Nb8 14.Ne2 transposes to Masisch-Westerinen,Senior Men world championship vallee d'Aoste 2006
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Now some answers to questions:
"When exactly to redeploy the a6N (after Bxa6) via b8...? "
Later is better, usually when you are forced to play this move...you need to have in your middlegame plan, safety of king as one main goal..and usually you cannot put it on kingside (because White attacks there) and not safe on center when center is not closed, then long castle is quickest way...that's why you see Bg5 often as a White move.
"When to move it forth to c6 (which is preceded or not by ..c5) and then to a5 ?"
Previous answer and analysis above give a clue: using knight to keep a blockade on light squares takes time, usually knight is well placed on a6 (where he has choice between Nb8 and Nc7)...whereas blockading with queen is easier with Qc7-Qa4/Qc6-Qc4. My answer is: usually your opponent will not let you do it, better to have two
ideas (use queen and use knight) than only one.
"When is the right moment to move the a8R (to c8)?"
It's part of the "what about my king's safety ?" question. White will have soon an attack on f and g files, can your king stay safely long time on e8 or f8 squares ? If king safety is not a problem, for example reaching a semi-ending usausally by trading queens, then c file is famously known (cf French pawn structures) as an excellent file for Black: for attacked squares and pawns...then the "moment" is "near the ending" or "not so far from queens removal off the board"
"When to castle long and when to short...?"
I prefer to castle "quickly"...

... If White plays regular sleepy moves as Nf3-Bd3-oo then after Ba6 and a bishops trade, I believe kingside catsling will be available...otherwise, you will to have to engineer long castle or maybe a "manual" castle (with Kd7).
"When is the queen shortie to a4 (via d7) sensible...? When to play for the exchange of it by ..Qd8-c8-a6 ?"
Those who have the "queen blues" will say...quite early (queen is a wonderful blockading piece)...but I will say sometimes queen is useful to protect f7 and g7 pawns..and help a manual castle.
"When is it correct to play for ..f6?"
System b6 is special, and undermining pawn center with f6 is clearly unavailable in many lines.
Don't be shy, you can post your games..