General Cognitive Dissonances About Opening Puzzles

Is my presumption correct that Stockfish is analyzing every move on all puzzles in real-time?

If so, the "best moves" really are the best moves and any questions arising about why a particular move is a best one and why another one makes no sense are due to lack of chess experience.

I frequently struggle with understanding why a particular move makes sense. For example pawn to a5 in the opening number 603: http://en.lichess.org/training/opening/603

I understand when one moves them in order to prevent an opponent's knight from forking, but in this particular opening puzzle like in many others that's not the case yet pawn to a5 is still a best move.

Does anybody have an explanation for why this is a best move?

I don't think the openings are being analyzed in real-time, but rather have already been analyzed by Stockfish and the best moves determined.

I assume you mean a4 for white. a4 is strong in this position simply because it threatens a5 which would kick black's knight to an unfavorable spot on d7 where it would further inhibit black's development. After a4 a5, white can play things like Nb5 to exploit black's lack of development. Furthermore, white can continue pushing the pawn to a6 to either activate his rook or create a thorn in black's queen side depending on how he reacts. Remember, the opening is all about development!

The "best moves" in the opening trainer focuses on analyzing most candidate moves and their possible lines and respective position centipawn. cp is the measurement of how a move changes the advantage, where a centipawn is 1/100 of a pawn and making a suboptimal move means you lose some centipawn.

By scrapping the server response of the puzzle, here's the analyzed information about the candidate moves in this puzzle:

Nb5: Identified as best move with centipawn -18.
Line continuation: Nb5 Ke7 O-O-O Bd7 Nd4 Rag8 Bd3 Na4 c3 Nc5 Bc2 g6 Kb1 b6 Rhe1 Rc8 g3 h5
a4: Identified as best move with centipawn -29.
Line continuation: a4 Ke7 a5 Nd7 Kd2 f6 exf6+ Nxf6 Bd3 Bd7 Ne2 Rhg8 h4 Raf8 Rhe1 Nh5 g3 g6 Rg1 Kd6 Rae1 Kc5 Nc3 Nf6 Rgf1
b3: Identified as best move with centipawn -31.
Line continuation: b3 Bd7 O-O-O Ke7
Kd2: Identified as best move with centipawn -23.
Line continuation: Kd2 Bd7 a4 a6 Bd3 Nc8 Ne2 Ne7 c3 Nc6 Nd4 Ke7 Nxc6+ Bxc6 a5 h5 Ke3 h4 Rhd1 d4+ cxd4 Bxg2 Rdc1
O-O-O: Identified as best move with centipawn -29.
Line continuation: O-O-O Bd7 b3 Rc8 Kb2 Ke7 Be2 Rc5 Rhf1 Rhc8 Rf3 a5 Rg3 a4 Rdd3 axb3 axb3 Bb5
Rg1: Identified as best move with centipawn -8.
Line continuation: Rg1 Bd7 Kd2 O-O a4 a6 a5 Nc8 Na4 Bxa4 Rxa4 Ne7 c3 Nc6 Be2 Rac8 g3 g6 Raa1 Kg7 Rgf1 f6 exf6+
Ne2: Identified as best move with centipawn -6.
Line continuation: Ne2 f6 Nd4 fxe5 fxe5 Ke7 O-O-O Nd7 Re1 Nc5 Bd3 Bd7 Kd2 Rac8 Rhf1 g6 b3 Rhf8 c4 b6 cxd5 exd5 g3 Ne6 Rxf8
g3: Identified as best move with centipawn -24.
Line continuation: g3 Bd7 a4 Rc8 Kd2
Be2: Identified as best move with centipawn -15.
Line continuation: Be2 Bd7 b3 Rc8 Kd2 f6 exf6 gxf6 a4 a5 g3 Ke7 Rhe1 h5 Bf3 h4 Nxd5+ Nxd5 Bxd5
a3: Identified as best move with centipawn -15.
Line continuation: a3 Bd7 b3 Rc8 Kd2 Ke7 a4 h6 Nb5 Bxb5 axb5 Ra8 Be2 Rhc8 g3 Nd7 Rhe1 g5 c4 gxf4 gxf4 Nc5 Kc2 Ne4
Bd3: Identified as best move with centipawn -20.
Line continuation: Bd3 Ke7 O-O-O Nd7 Be2 b6 Bf3 Bb7 Nb5 g5 g3 Bc6 Nd4 Rac8 Kb1 Bb7 Rhg1 Rhg8
g4: Identified as suboptimal move with centipawn 0.
Line continuation: g4 h5 g5 Bd7 O-O-O Rc8 h4 Ke7 Be2 a6 Rh3 Na4 f5 Nxc3 f6+ gxf6 gxf6+ Kd8 Rxc3 Rxc3 bxc3 Kc7 c4 dxc4 Bxc4
Rd1: Identified as suboptimal move with centipawn 1.
Line continuation: Rd1 Bd7 Be2 Rc8 Rd4 f6 exf6 gxf6 O-O Kf7 Rb4 Rc5 a4 Rhc8 Ra1 Ra5 g3 Kg7
h4: Identified as suboptimal move with centipawn 4.
Line continuation: h4 Bd7 O-O-O Rc8 Kb1 O-O g3 Nc4 Rd3 b5 Ne2 b4 Nd4 a5 Bg2 a4 c3 Rb8
Bb5+: Identified as suboptimal move with centipawn 5.
Line continuation: Bb5+ Bd7 O-O-O Ke7 Rhe1 h6 Be2 Rac8 Rd4 Nc4 Bxc4 Rxc4 Rxc4 dxc4 g3 Bc6 Rd1 Rc8 a3 f6 Rd4 b5 Rd6 a6 Kd2 fxe5
h3: Identified as suboptimal move with centipawn 4.
Line continuation: h3 Bd7 a4 Rc8 Kd2 Nc4+ Bxc4 Rxc4 Ne2 Ke7 b3 Re4 Rhf1 Rc8 Kd3
Rc1: Identified as suboptimal move with centipawn 8.
Line continuation: Rc1 Bd7 Kf2 Ke7 Bd3 Rac8 Ne2 Na4 Rb1 Nc5 Nd4 h6 h4 a6 g3 Nxd3+
Rb1: Identified as suboptimal move with centipawn 6.
Line continuation: Rb1 Bd7 Kd2 Rd8 Bd3 Bc6 Ne2 d4 Rhf1 Ke7 g3 h5 h4 g6 b3 Nd7
Ke2: Identified as suboptimal move with centipawn 7.
Line continuation: Ke2 Bd7 Kd2 Bc6 a4 O-O a5 d4 axb6 dxc3+ Kxc3 axb6 Bd3 Bxg2 Rhg1 Rfc8+ Kd4 Rxa1 Rxa1 h6 Rg1 Rd8+ Ke3 Bc6
Kf2: Identified as suboptimal move with centipawn 10.
Line continuation: Kf2 Bd7 a4 Rc8 Bd3 O-O Nb5 Nxa4 Nd6 Ra8 b3 Nc5 b4 Nxd3+ cxd3 f6 exf6

I think the rest are considered poor moves.