Move order

I put the first position in the computer and the difference is minisule (d6 gives -0.08 and Bxe3 gives 0.00, which is basically the same). It also likes Qe8.

Basically the comp eval is "slight advantage for black (comp. returns more favorable lines for black). White draw with best play". Therefore, Bxe3 does trade the piece rather prematurely and somewhat help white in getting a draw. Also notice that the comp. thinks Bxb6? is a mistake for white in most lines and gives like a -0.4 after axb6, so in essence only black can initiate the trade and black can do it almost anytime. So not trading does have the benefit of keeping the tension and keeping white's queen around the bishop.

After d6, Qd2?! is not a best move for the simple reason that it does nothing except simply waste a tempo. This is a good time to take being a tempo up (white hasn't played Kb1 for example) and obviously it is better than the tempo down version.

Good answer. However, I'm still quite confused. Why is it better to take the bishop after you have gained a tempo due to whites Qd2? After all, if I just look at the position without any knowledge of the previous mpves made, I should certainly come to the same conclusion about the best next move for black without knowing that white just played Qd2. So why not postpone the capture and keep the tension here as well? Usually in the opening, if you are tempos up you don't want to trade.

Personally I would play d6 in both positions. 

The way tempo is considered here is whether white have other useful moves that the tempo might be important. In some positions the tempo doesn't really matter. In the position given though I do see one necessary move for white: Kb1 is necessary in almost all situations. Therefore the difference in the two charts after Bxe3 is that the white king is on c1 instead of b1 (the lost tempo)

It is very hard to imagine white not playing Kb1 at some point, either immediately or maybe 5-10 moves later. The problem with the king on c1 is it could be a tactics liability (a piece could be pinned to the king) or a tempo gainer for black (with a check). This tempo does matter in this open position as white wants to play actively to try to attack black's defense, and it is harder to do that with the king not in the castle. And if white plays Kb1 that is obviously not an active move and black gets in an extra defensive move.

Therefore, Qd2?! is not a good move because it took the tempo away from Kb1, and thus Kb1! is the recommended move.

With Qd2?! the comp realizes that the maximum effect is reached and black does need to take the bishop to realize the advantage. If black plays like a nothing move (like a6) or some other weak move, white is going Bc4+ (note that the queen got out of the way so the bishop can come out), Kh8 and Re1 with clear advantage.